Lista dé integráls dé fonczións trigonométrica

Cania es et lista dé integráls (fonczións eźederivativa) dé fonczións trigonométrica. Dés eźederivativas involver carna fonczións exponentiál én trigonométrica, nior Lista dé integráls dé fonczións exponentiál. Dés et lista hetacaró dé fonczións eźederivativa, nior Tabla dé integráls. Dés ét eźederivativas especiál involver fonczións trigonométrica, nior Integrál trigonométrica.

Generelamentá, né ét fonczión

sin
⁡
x

{\displaystyle \sin x}

 es casié fonczión trigonométrica, én

cos
⁡
x

{\displaystyle \cos x}

 es bous derivativa,

∫
a
cos
⁡
n
x

d
# x

a
n

sin
⁡
n
x
C

{\displaystyle \int a\cos nx\,dx = \frac{a}{n}\sin nx+C}

Ens tace formulas ét constanta *a* es nader a ser imcero, én *C* denotar ét constanta dé integratazión.

## Integráls involver mós **sín**

* ∫
sin
⁡
a
x

d
# x
−

1
a

cos
⁡
a
x
C

{\displaystyle \int\sin ax\,dx = -\frac{1}{a}\cos ax+C}
* ∫

sin
2

⁡

a
x

d
# x

x
2

−

1

4
a

sin
⁡
2
a
x
# C

x
2

−

1

2
a

sin
⁡
a
x
cos
⁡
a
x
C

{\displaystyle \int\sin^2 {ax}\,dx = \frac{x}{2} - \frac{1}{4a} \sin 2ax +C= \frac{x}{2} - \frac{1}{2a} \sin ax\cos ax +C}
* ∫

sin
3

⁡

a
x

d
# x

cos
⁡
3
a
x

12
a

−

3
cos
⁡
a
x

4
a

C

{\displaystyle \int\sin^3 {ax}\,dx = \frac{\cos 3ax}{12a} - \frac{3 \cos ax}{4a} +C}
* ∫
x

sin
2

⁡

a
x

d
# x

x
2

4

−

x

4
a

sin
⁡
2
a
x
−

1

8

a
2

cos
⁡
2
a
x
C

{\displaystyle \int x\sin^2 {ax}\,dx = \frac{x^2}{4} - \frac{x}{4a} \sin 2ax - \frac{1}{8a^2} \cos 2ax +C}
* ∫

x
2

sin
2

⁡

a
x

d
# x

x
3

6

−

(

x
2

4
a

−

1

8

a
3

)

sin
⁡
2
a
x
−

x

4

a
2

cos
⁡
2
a
x
C

{\displaystyle \int x^2\sin^2 {ax}\,dx = \frac{x^3}{6} - \left( \frac {x^2}{4a} - \frac{1}{8a^3} \right) \sin 2ax - \frac{x}{4a^2} \cos 2ax +C}
* ∫
x
sin
⁡
a
x

d
# x

sin
⁡
a
x

a
2

−

x
cos
⁡
a
x

a

C

{\displaystyle \int x\sin ax\,dx = \frac{\sin ax}{a^2}-\frac{x\cos ax}{a}+C}
* ∫
(
sin
⁡

b
1

x
)
(
sin
⁡

b
2

x
)

d
# x

sin
⁡
(
(

b
2

−

b
1

)
x
)

2
(

b
2

−

b
1

)

−

sin
⁡
(
(

b
1

b
2

)
x
)

2
(

b
1

b
2

)

C

(for

b
1

≠

b
2

)

* ∫

sin
n

⁡

a
x

d
# x
−

sin

n
−
1

⁡
a
x
cos
⁡
a
x

n
a

n
−
1

n

∫

sin

n
−
2

⁡
a
x

d
x

(for

n
>
0

)

{\displaystyle \int\sin^n {ax}\,dx = -\frac{\sin^{n-1} ax\cos ax}{na} + \frac{n-1}{n}\int\sin^{n-2} ax\,dx \qquad\mbox{(for }n>0\mbox{)}}
* ∫

d
x

sin
⁡
a
x

=
−

1
a

ln
⁡

csc
⁡

a
x

cot
⁡

a
x

C

{\displaystyle \int\frac{dx}{\sin ax} = -\frac{1}{a}\ln{\left| \csc{ax}+\cot{ax}\right|}+C}
* ∫

d
x

sin
n

⁡
a
x

=

cos
⁡
a
x

a
(
1
−
n
)

sin

n
−
1

⁡
a
x

n
−
2

n
−
1

∫

d
x

sin

n
−
2

⁡
a
x

(for

n
>
1

)

{\displaystyle \int\frac{dx}{\sin^n ax} = \frac{\cos ax}{a(1-n) \sin^{n-1} ax}+\frac{n-2}{n-1}\int\frac{dx}{\sin^{n-2}ax} \qquad\mbox{(for }n>1\mbox{)}}
* ∫

x
n

sin
⁡
a
x

d
x

=
−

x
n

a

cos
⁡
a
x

n
a

∫

x

n
−
1

cos
⁡
a
x

d
x

=

∑

# k
0

2
k
≤
n

(
−
1

)

k
1

x

n
−
2
k

a

1
2
k

n
!

(
n
−
2
k
)
!

cos
⁡
a
x

∑

# k
0

2
k
1
≤
n

(
−
1

)
k

x

n
−
1
−
2
k

a

2
2
k

n
!

(
n
−
2
k
−
1
)
!

sin
⁡
a
x

=
−

∑

# k
0

n

x

n
−
k

a

1
k

n
!

(
n
−
k
)
!

cos
⁡

(

a
x
k

π
2

)

(for

n
>
0

)

{\displaystyle \begin{align}
\int x^n\sin ax\,dx &= -\frac{x^n}{a}\cos ax+\frac{n}{a}\int x^{n-1}\cos ax\,dx \\
&= \sum\_{k=0}^{2k\leq n} (-1)^{k+1} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \cos ax +\sum\_{k=0}^{2k+1\leq n}(-1)^k \frac{x^{n-1-2k}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \sin ax \\
&= - \sum\_{k=0}^n \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\cos\left(ax+k\frac{\pi}{2}\right) \qquad\mbox{(for }n>0\mbox{)}
\end{align}}
* ∫

sin
⁡
a
x

x

d

# n
0

∞

(
−
1

)
n

(
a
x

)

2
n
1

(
2
n
1
)
⋅
(
2
n
1
)
!

C

{\displaystyle \int\frac{\sin ax}{x}\,dx = \sum\_{n=0}^\infty (-1)^n\frac{(ax)^{2n+1}}{(2n+1)\cdot (2n+1)!} +C}
* ∫

sin
⁡
a
x

x
n

d
# x
−

sin
⁡
a
x

(
n
−
1
)

x

n
−
1

a

n
−
1

∫

cos
⁡
a
x

x

n
−
1

d
x

{\displaystyle \int\frac{\sin ax}{x^n}\,dx = -\frac{\sin ax}{(n-1)x^{n-1}} + \frac{a}{n-1}\int\frac{\cos ax}{x^{n-1}}\,dx}
* ∫

sin
⁡

(

a
x

2

b
x

c

)

d
x

=

{

a

π

2

cos
⁡

(

b

2

−

4

a
c

4
a

)

S

(

2

a
x

b

2

a

π

)

a

π

2

sin
⁡

(

b

2

−

4

a
c

4
a

)

C

(

2

a
x

b

2

a

π

)

t
o

b

2

−

4

a
c

>

0

a

π

2

cos
⁡

(

b

2

−

4

a
c

4
a

)

S

(

2

a
x

b

2

a

π

)

−

a

π

2

sin
⁡

(

b

2

−

4

a
c

4
a

)

C

(

2

a
x

b

2

a

π

)

t
o

b

2

−

4

a
c

<

0

f
o
r

a

╱

=

0

,

a

>

0

{\displaystyle
\int{\sin{\mathrm{(}}{ax}^{2}\mathrm{{+}}{bx}\mathrm{{+}}{c}{\mathrm{)}}{dx}}\mathrm{{=}}\left\{{\begin{align}
& {\sqrt{a}\sqrt{\frac{\mathit{\pi}}{2}}\cos\left({\frac{{b}^{2}\mathrm{{-}}{4}{ac}}{4a}}\right){S}\left({\frac{{2}{ax}\mathrm{{+}}{b}}{\sqrt{{2}{a}\mathit{\pi}}}}\right)\mathrm{{+}}\sqrt{a}\sqrt{\frac{\mathit{\pi}}{2}}\sin\left({\frac{{b}^{2}\mathrm{{-}}{4}{ac}}{4a}}\right){C}\left({\frac{{2}{ax}\mathrm{{+}}{b}}{\sqrt{{2}{a}\mathit{\pi}}}}\right)\;{to}\;{b}^{2}\mathrm{{-}}{4}{ac}\;{\mathrm{>}}\;{0}}\\
& {\sqrt{a}\sqrt{\frac{\mathit{\pi}}{2}}\cos\left({\frac{{b}^{2}\mathrm{{-}}{4}{ac}}{4a}}\right){S}\left({\frac{{2}{ax}\mathrm{{+}}{b}}{\sqrt{{2}{a}\mathit{\pi}}}}\right)\mathrm{{-}}\sqrt{a}\sqrt{\frac{\mathit{\pi}}{2}}\sin\left({\frac{{b}^{2}\mathrm{{-}}{4}{ac}}{4a}}\right){C}\left({\frac{{2}{ax}\mathrm{{+}}{b}}{\sqrt{{2}{a}\mathit{\pi}}}}\right)\;{to}\;{b}^{2}\mathrm{{-}}{4}{ac}\;{\mathrm{<}}\;{0}}
\end{align}}\right.\;\;{for}\;{a}\diagup\!\!\!\!{\mathrm{{=}}}{0}{\mathrm{,}}\;{a}{\mathrm{>}}{0}
}
* ∫

d
x

1
±
sin
⁡
a
x

=

1
a

tan
⁡

(

a
x

2

∓

π
4

)

C

{\displaystyle \int\frac{dx}{1\pm\sin ax} = \frac{1}{a}\tan\left(\frac{ax}{2}\mp\frac{\pi}{4}\right)+C}
* ∫

x

d
x

1
sin
⁡
a
x

=

x
a

tan
⁡

(

a
x

2

−

π
4

)

2

a
2

ln
⁡

cos
⁡

(

a
x

2

−

π
4

)

C

{\displaystyle \int\frac{x\,dx}{1+\sin ax} = \frac{x}{a}\tan\left(\frac{ax}{2} - \frac{\pi}{4}\right)+\frac{2}{a^2}\ln\left|\cos\left(\frac{ax}{2}-\frac{\pi}{4}\right)\right|+C}
* ∫

x

d
x

1
−
sin
⁡
a
x

=

x
a

cot
⁡

(

π
4

−

a
x

2

)

2

a
2

ln
⁡

sin
⁡

(

π
4

−

a
x

2

)

C

{\displaystyle \int\frac{x\,dx}{1-\sin ax} = \frac{x}{a}\cot\left(\frac{\pi}{4} - \frac{ax}{2}\right)+\frac{2}{a^2}\ln\left|\sin\left(\frac{\pi}{4}-\frac{ax}{2}\right)\right|+C}
* ∫

sin
⁡
a
x

d
x

1
±
sin
⁡
a
x

=
±
x

1
a

tan
⁡

(

π
4

∓

a
x

2

)

C

{\displaystyle \int\frac{\sin ax\,dx}{1\pm\sin ax} = \pm x+\frac{1}{a}\tan\left(\frac{\pi}{4}\mp\frac{ax}{2}\right)+C}

## Integráls involver mós **cosín**

* ∫
cos
⁡
a
x

d
# x

1
a

sin
⁡
a
x
C

{\displaystyle \int\cos ax\,dx = \frac{1}{a}\sin ax+C}
* ∫

cos
2

⁡

a
x

d
# x

x
2

1

4
a

sin
⁡
2
a
x
# C

x
2

1

2
a

sin
⁡
a
x
cos
⁡
a
x
C

{\displaystyle \int\cos^2 {ax}\,dx = \frac{x}{2} + \frac{1}{4a} \sin 2ax +C = \frac{x}{2} + \frac{1}{2a} \sin ax\cos ax +C}
* ∫

cos
n

⁡
a
x

d
# x

cos

n
−
1

⁡
a
x
sin
⁡
a
x

n
a

n
−
1

n

∫

cos

n
−
2

⁡
a
x

d
x

(for

n
>
0

)

{\displaystyle \int\cos^n ax\,dx = \frac{\cos^{n-1} ax\sin ax}{na} + \frac{n-1}{n}\int\cos^{n-2} ax\,dx \qquad\mbox{(for }n>0\mbox{)}}
* ∫
x
cos
⁡
a
x

d
# x

cos
⁡
a
x

a
2

x
sin
⁡
a
x

a

C

{\displaystyle \int x\cos ax\,dx = \frac{\cos ax}{a^2} + \frac{x\sin ax}{a}+C}
* ∫

x
2

cos
2

⁡

a
x

d
# x

x
3

6

(

x
2

4
a

−

1

8

a
3

)

sin
⁡
2
a
x

x

4

a
2

cos
⁡
2
a
x
C

{\displaystyle \int x^2\cos^2 {ax}\,dx = \frac{x^3}{6} + \left( \frac {x^2}{4a} - \frac{1}{8a^3} \right) \sin 2ax + \frac{x}{4a^2} \cos 2ax +C}
* ∫

x
n

cos
⁡
a
x

d
x

=

x
n

sin
⁡
a
x

a

−

n
a

∫

x

n
−
1

sin
⁡
a
x

d
x

=

∑

# k
0

2
k
1
≤
n

(
−
1

)

k

x

n
−
2
k
−
1

a

2
2
k

n
!

(
n
−
2
k
−
1
)
!

cos
⁡
a
x

∑

# k
0

2
k
≤
n

(
−
1

)

k

x

n
−
2
k

a

1
2
k

n
!

(
n
−
2
k
)
!

sin
⁡
a
x

=

∑

# k
0

n

(
−
1

)

⌊
k

/

2
⌋

x

n
−
k

a

1
k

n
!

(
n
−
k
)
!

cos
⁡

(

a
x
−

(
−
1

)
k

1

2

π
2

)

=

∑

# k
0

n

x

n
−
k

a

1
k

n
!

(
n
−
k
)
!

sin
⁡

(

a
x
k

π
2

)

(for

n
>
0

)

{\displaystyle \begin{align}
\int x^n\cos ax\,dx &= \frac{x^n\sin ax}{a} - \frac{n}{a}\int x^{n-1}\sin ax\,dx \\
&= \sum\_{k=0}^{2k+1\leq n} (-1)^{k} \frac{x^{n-2k-1}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \cos ax +\sum\_{k=0}^{2k\leq n}(-1)^{k} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \sin ax \\
&=\sum\_{k=0}^n (-1)^{\lfloor k/2 \rfloor} \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\cos\left(ax -\frac{(-1)^k+1}{2}\frac{\pi}{2}\right) \\
&=\sum\_{k=0}^n \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\sin\left(ax+k\frac{\pi}{2}\right) \qquad\mbox{(for }n>0\mbox{)}
\end{align}}
* ∫

cos
⁡
a
x

x

d
# x
ln
⁡

a
x

∑

# k
1

∞

(
−
1

)
k

(
a
x

)

2
k

2
k
⋅
(
2
k
)
!

C

{\displaystyle \int\frac{\cos ax}{x}\,dx = \ln|ax|+\sum\_{k=1}^\infty (-1)^k\frac{(ax)^{2k}}{2k\cdot(2k)!}+C}
* ∫

cos
⁡
a
x

x
n

d
# x
−

cos
⁡
a
x

(
n
−
1
)

x

n
−
1

−

a

n
−
1

∫

sin
⁡
a
x

x

n
−
1

d
x

(for

n
≠
1

)

{\displaystyle \int\frac{\cos ax}{x^n}\,dx = -\frac{\cos ax}{(n-1)x^{n-1}}-\frac{a}{n-1}\int\frac{\sin ax}{x^{n-1}}\,dx \qquad\mbox{(for }n\neq 1\mbox{)}}
* ∫

d
x

cos
⁡
a
x

=

1
a

ln
⁡

tan
⁡

(

a
x

2

π
4

)

C

{\displaystyle \int\frac{dx}{\cos ax} = \frac{1}{a}\ln\left|\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right)\right|+C}
* ∫

d
x

cos
n

⁡
a
x

=

sin
⁡
a
x

a
(
n
−
1
)

cos

n
−
1

⁡
a
x

n
−
2

n
−
1

∫

d
x

cos

n
−
2

⁡
a
x

(for

n
>
1

)

{\displaystyle \int\frac{dx}{\cos^n ax} = \frac{\sin ax}{a(n-1) \cos^{n-1} ax} + \frac{n-2}{n-1}\int\frac{dx}{\cos^{n-2} ax} \qquad\mbox{(for }n>1\mbox{)}}
* ∫

d
x

1
cos
⁡
a
x

=

1
a

tan
⁡

a
x

2

C

{\displaystyle \int\frac{dx}{1+\cos ax} = \frac{1}{a}\tan\frac{ax}{2}+C}
* ∫

d
x

1
−
cos
⁡
a
x

=
−

1
a

cot
⁡

a
x

2

C

{\displaystyle \int\frac{dx}{1-\cos ax} = -\frac{1}{a}\cot\frac{ax}{2}+C}
* ∫

x

d
x

1
cos
⁡
a
x

=

x
a

tan
⁡

a
x

2

2

a
2

ln
⁡

cos
⁡

a
x

2

C

{\displaystyle \int\frac{x\,dx}{1+\cos ax} = \frac{x}{a}\tan\frac{ax}{2} + \frac{2}{a^2}\ln\left|\cos\frac{ax}{2}\right|+C}
* ∫

x

d
x

1
−
cos
⁡
a
x

=
−

x
a

cot
⁡

a
x

2

2

a
2

ln
⁡

sin
⁡

a
x

2

C

{\displaystyle \int\frac{x\,dx}{1-\cos ax} = -\frac{x}{a}\cot\frac{ax}{2}+\frac{2}{a^2}\ln\left|\sin\frac{ax}{2}\right|+C}
* ∫

cos
⁡
a
x

d
x

1
cos
⁡
a
x

=
x
−

1
a

tan
⁡

a
x

2

C

{\displaystyle \int\frac{\cos ax\,dx}{1+\cos ax} = x - \frac{1}{a}\tan\frac{ax}{2}+C}
* ∫

cos
⁡
a
x

d
x

1
−
cos
⁡
a
x

=
−
x
−

1
a

cot
⁡

a
x

2

C

{\displaystyle \int\frac{\cos ax\,dx}{1-\cos ax} = -x-\frac{1}{a}\cot\frac{ax}{2}+C}
* ∫
(
cos
⁡

a
1

x
)
(
cos
⁡

a
2

x
)

d
# x

sin
⁡
(
(

a
2

−

a
1

)
x
)

2
(

a
2

−

a
1

)

sin
⁡
(
(

a
2

a
1

)
x
)

2
(

a
2

a
1

)

C

(for

a
1

≠

a
2

)

## Integráls involver mós **tangenta**

* ∫
tan
⁡
a
x

d
# x
−

1
a

ln
⁡

cos
⁡
a
x

# C

1
a

ln
⁡

sec
⁡
a
x

C

* ∫

tan
2

⁡

x

d
# x
tan
⁡

x

−
x
C

{\displaystyle \int \tan^2{x} \, dx = \tan{x} - x +C}
* ∫

tan
n

⁡
a
x

d
# x

1

a
(
n
−
1
)

tan

n
−
1

⁡
a
x
−
∫

tan

n
−
2

⁡
a
x

d
x

(for

n
≠
1

)

{\displaystyle \int\tan^n ax\,dx = \frac{1}{a(n-1)}\tan^{n-1} ax-\int\tan^{n-2} ax\,dx \qquad\mbox{(for }n\neq 1\mbox{)}}
* ∫

d
x

q
tan
⁡
a
x
p

=

1

p
2

q
2

(
p
x

q
a

ln
⁡

q
sin
⁡
a
x
p
cos
⁡
a
x

)
C

(for

p
2

q
2

≠
0

)

{\displaystyle \int\frac{dx}{q \tan ax + p} = \frac{1}{p^2 + q^2}(px + \frac{q}{a}\ln|q\sin ax + p\cos ax|)+C \qquad\mbox{(for }p^2 + q^2\neq 0\mbox{)}}
* ∫

d
x

tan
⁡
a
x
±
1

=
±

x
2

1

2
a

ln
⁡

sin
⁡
a
x
±
cos
⁡
a
x

C

{\displaystyle \int\frac{dx}{\tan ax \pm 1} = \pm \frac{x}{2} + \frac{1}{2a}\ln|\sin ax \pm \cos ax|+C}
* ∫

tan
⁡
a
x

d
x

tan
⁡
a
x
±
1

=

x
2

∓

1

2
a

ln
⁡

sin
⁡
a
x
±
cos
⁡
a
x

C

{\displaystyle \int\frac{\tan ax\,dx}{\tan ax \pm 1} = \frac{x}{2} \mp \frac{1}{2a}\ln|\sin ax \pm \cos ax|+C}

## Integráls involver mós **cotangenta**

* ∫
cot
⁡
a
x

d
# x

1
a

ln
⁡

sin
⁡
a
x

C

{\displaystyle \int\cot ax\,dx = \frac{1}{a}\ln|\sin ax|+C}
* ∫

cot
2

⁡

x

d
# x
−
cot
⁡

x

−
x
C

{\displaystyle \int \cot^2{x} \, dx = -\cot{x} - x +C}
* ∫

cot
n

⁡
a
x

d
# x
−

1

a
(
n
−
1
)

cot

n
−
1

⁡
a
x
−
∫

cot

n
−
2

⁡
a
x

d
x

(for

n
≠
1

)

{\displaystyle \int\cot^n ax\,dx = -\frac{1}{a(n-1)}\cot^{n-1} ax - \int\cot^{n-2} ax\,dx \qquad\mbox{(for }n\neq 1\mbox{)}}
* ∫

d
x

1
cot
⁡
a
x

=
∫

tan
⁡
a
x

d
x

tan
⁡
a
x
1

=

x
2

−

1

2
a

ln
⁡

sin
⁡
a
x
cos
⁡
a
x

C

{\displaystyle \int\frac{dx}{1 + \cot ax} = \int\frac{\tan ax\,dx}{\tan ax+1} = \frac{x}{2} - \frac{1}{2a}\ln|\sin ax + \cos ax|+C }
* ∫

d
x

1
−
cot
⁡
a
x

=
∫

tan
⁡
a
x

d
x

tan
⁡
a
x
−
1

=

x
2

1

2
a

ln
⁡

sin
⁡
a
x
−
cos
⁡
a
x

C

{\displaystyle \int\frac{dx}{1 - \cot ax} = \int\frac{\tan ax\,dx}{\tan ax-1} = \frac{x}{2} + \frac{1}{2a}\ln|\sin ax - \cos ax|+C }

## Integráls involver mós **secanta**

* ∫
sec
⁡

a
x

d
# x

1
a

ln
⁡

sec
⁡

a
x

tan
⁡

a
x

# C

1
a

ln
⁡

tan
⁡

(

a
x

2

π
4

)

# C

1
a

artanh
⁡

(

sin
⁡

a
x

)

C

* ∫

sec
2

⁡

x

d
# x
tan
⁡

x

C

{\displaystyle \int \sec^2{x} \, dx = \tan{x}+C}
* ∫

sec
3

⁡

x

d
# x

1
2

sec
⁡
x
tan
⁡
x

1
2

ln
⁡

sec
⁡
x
tan
⁡
x

C
.

{\displaystyle \int \sec^3{x} \, dx = \frac{1}{2}\sec x \tan x + \frac{1}{2}\ln|\sec x + \tan x| + C.}
* ∫

sec
n

⁡

a
x

d
# x

sec

n
−
2

⁡

a
x

tan
⁡

a
x

a
(
n
−
1
)

n
−
2

n
−
1

∫

sec

n
−
2

⁡

a
x

d
x

 (for

n
≠
1

)

{\displaystyle \int \sec^n{ax} \, dx = \frac{\sec^{n-2}{ax} \tan {ax}}{a(n-1)} \,+\, \frac{n-2}{n-1}\int \sec^{n-2}{ax} \, dx \qquad \mbox{ (for }n \ne 1\mbox{)}}
* ∫

d
x

sec
⁡

x

1

=
x
−
tan
⁡

x
2

C

{\displaystyle \int \frac{dx}{\sec{x} + 1} = x - \tan{\frac{x}{2}}+C}
* ∫

d
x

sec
⁡

x

−
1

=
−
x
−
cot
⁡

x
2

C

{\displaystyle \int \frac{dx}{\sec{x} - 1} = - x - \cot{\frac{x}{2}}+C}

## Integráls involver mós **cosecanta**

* ∫
csc
⁡

a
x

d
# x
−

1
a

ln
⁡

csc
⁡

a
x

cot
⁡

a
x

# C

1
a

ln
⁡

csc
⁡

a
x

−
cot
⁡

a
x

# C

1
a

ln
⁡

tan
⁡

(

a
x

2

)

C

* ∫

csc
2

⁡

x

d
# x
−
cot
⁡

x

C

{\displaystyle \int \csc^2{x} \, dx = -\cot{x}+C}
* ∫

csc
3

⁡

x

d
# x
−

1
2

csc
⁡
x
cot
⁡
x
−

1
2

ln
⁡

csc
⁡
x
cot
⁡
x

# C
−

1
2

csc
⁡
x
cot
⁡
x

1
2

ln
⁡

csc
⁡
x
−
cot
⁡
x

C

* ∫

csc
n

⁡

a
x

d
# x
−

csc

n
−
2

⁡

a
x

cot
⁡

a
x

a
(
n
−
1
)

n
−
2

n
−
1

∫

csc

n
−
2

⁡

a
x

d
x

 (for

n
≠
1

)

{\displaystyle \int \csc^n{ax} \, dx = -\frac{\csc^{n-2}{ax} \cot{ax}}{a(n-1)} \,+\, \frac{n-2}{n-1}\int \csc^{n-2}{ax} \, dx \qquad \mbox{ (for }n \ne 1\mbox{)}}
* ∫

d
x

csc
⁡

x

1

=
x
−

2

cot
⁡

x
2

1

C

{\displaystyle \int \frac{dx}{\csc{x} + 1} = x - \frac{2}{\cot{\frac{x}{2}}+1}+C}
* ∫

d
x

csc
⁡

x

−
1

=
−
x

2

cot
⁡

x
2

−
1

C

{\displaystyle \int \frac{dx}{\csc{x} - 1} = - x + \frac{2}{\cot{\frac{x}{2}}-1}+C}

## Integráls involver carna **sín** én **cosín**

Et integrál quil es et fonczión razionál dé ét sín én cosín ril ser evalar, usir luźos dé Bioche.

* ∫

d
x

cos
⁡
a
x
±
sin
⁡
a
x

=

1

a

2

ln
⁡

tan
⁡

(

a
x

2

±

π
8

)

C

{\displaystyle \int\frac{dx}{\cos ax\pm\sin ax} = \frac{1}{a\sqrt{2}}\ln\left|\tan\left(\frac{ax}{2}\pm\frac{\pi}{8}\right)\right|+C}
* ∫

d
x

(
cos
⁡
a
x
±
sin
⁡
a
x

)
2

=

1

2
a

tan
⁡

(

a
x
∓

π
4

)

C

{\displaystyle \int\frac{dx}{(\cos ax\pm\sin ax)^2} = \frac{1}{2a}\tan\left(ax\mp\frac{\pi}{4}\right)+C}
* ∫

d
x

(
cos
⁡
x
sin
⁡
x

)
n

=

1

2
(
n
−
1
)

(

sin
⁡
x
−
cos
⁡
x

(
cos
⁡
x
sin
⁡
x

)

n
−
1

(
n
−
2
)
∫

d
x

(
cos
⁡
x
sin
⁡
x

)

n
−
2

)

{\displaystyle \int\frac{dx}{(\cos x + \sin x)^n} = \frac{1}{2(n-1)}\left(\frac{\sin x - \cos x}{(\cos x + \sin x)^{n - 1}} + (n - 2)\int\frac{dx}{(\cos x + \sin x)^{n-2}} \right)}
* ∫

cos
⁡
a
x

d
x

cos
⁡
a
x
sin
⁡
a
x

=

x
2

1

2
a

ln
⁡

sin
⁡
a
x
cos
⁡
a
x

C

{\displaystyle \int\frac{\cos ax\,dx}{\cos ax + \sin ax} = \frac{x}{2} + \frac{1}{2a}\ln\left|\sin ax + \cos ax\right|+C}
* ∫

cos
⁡
a
x

d
x

cos
⁡
a
x
−
sin
⁡
a
x

=

x
2

−

1

2
a

ln
⁡

sin
⁡
a
x
−
cos
⁡
a
x

C

{\displaystyle \int\frac{\cos ax\,dx}{\cos ax - \sin ax} = \frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax - \cos ax\right|+C}
* ∫

sin
⁡
a
x

d
x

cos
⁡
a
x
sin
⁡
a
x

=

x
2

−

1

2
a

ln
⁡

sin
⁡
a
x
cos
⁡
a
x

C

{\displaystyle \int\frac{\sin ax\,dx}{\cos ax + \sin ax} = \frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax + \cos ax\right|+C}
* ∫

sin
⁡
a
x

d
x

cos
⁡
a
x
−
sin
⁡
a
x

=
−

x
2

−

1

2
a

ln
⁡

sin
⁡
a
x
−
cos
⁡
a
x

C

{\displaystyle \int\frac{\sin ax\,dx}{\cos ax - \sin ax} = -\frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax - \cos ax\right|+C}
* ∫

cos
⁡
a
x

d
x

(
sin
⁡
a
x
)
(
1
cos
⁡
a
x
)

=
−

1

4
a

tan
2

⁡

a
x

2

1

2
a

ln
⁡

tan
⁡

a
x

2

C

{\displaystyle \int\frac{\cos ax\,dx}{(\sin ax)(1+\cos ax)} = -\frac{1}{4a}\tan^2\frac{ax}{2}+\frac{1}{2a}\ln\left|\tan\frac{ax}{2}\right|+C}
* ∫

cos
⁡
a
x

d
x

(
sin
⁡
a
x
)
(
1
−
cos
⁡
a
x
)

=
−

1

4
a

cot
2

⁡

a
x

2

−

1

2
a

ln
⁡

tan
⁡

a
x

2

C

{\displaystyle \int\frac{\cos ax\,dx}{(\sin ax)(1-\cos ax)} = -\frac{1}{4a}\cot^2\frac{ax}{2}-\frac{1}{2a}\ln\left|\tan\frac{ax}{2}\right|+C}
* ∫

sin
⁡
a
x

d
x

(
cos
⁡
a
x
)
(
1
sin
⁡
a
x
)

=

1

4
a

cot
2

⁡

(

a
x

2

π
4

)

1

2
a

ln
⁡

tan
⁡

(

a
x

2

π
4

)

C

{\displaystyle \int\frac{\sin ax\,dx}{(\cos ax)(1+\sin ax)} = \frac{1}{4a}\cot^2\left(\frac{ax}{2}+\frac{\pi}{4}\right)+\frac{1}{2a}\ln\left|\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right)\right|+C}
* ∫

sin
⁡
a
x

d
x

(
cos
⁡
a
x
)
(
1
−
sin
⁡
a
x
)

=

1

4
a

tan
2

⁡

(

a
x

2

π
4

)

−

1

2
a

ln
⁡

tan
⁡

(

a
x

2

π
4

)

C

{\displaystyle \int\frac{\sin ax\,dx}{(\cos ax)(1-\sin ax)} = \frac{1}{4a}\tan^2\left(\frac{ax}{2}+\frac{\pi}{4}\right)-\frac{1}{2a}\ln\left|\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right)\right|+C}
* ∫
(
sin
⁡
a
x
)
(
cos
⁡
a
x
)

d
# x

1

2
a

sin
2

⁡
a
x
C

{\displaystyle \int(\sin ax)(\cos ax)\,dx = \frac{1}{2a}\sin^2 ax +C}
* ∫
(
sin
⁡

a
1

x
)
(
cos
⁡

a
2

x
)

d
# x
−

cos
⁡
(
(

a
1

−

a
2

)
x
)

2
(

a
1

−

a
2

)

−

cos
⁡
(
(

a
1

a
2

)
x
)

2
(

a
1

a
2

)

C

(for

a
1

≠

a
2

)

* ∫
(

sin
n

⁡
a
x
)
(
cos
⁡
a
x
)

d
# x

1

a
(
n
1
)

sin

n
1

⁡
a
x
C

(for

n
≠
−
1

)

{\displaystyle \int(\sin^n ax)(\cos ax)\,dx = \frac{1}{a(n+1)}\sin^{n+1} ax +C\qquad\mbox{(for }n\neq -1\mbox{)}}
* ∫
(
sin
⁡
a
x
)
(

cos
n

⁡
a
x
)

d
# x
−

1

a
(
n
1
)

cos

n
1

⁡
a
x
C

(for

n
≠
−
1

)

{\displaystyle \int(\sin ax)(\cos^n ax)\,dx = -\frac{1}{a(n+1)}\cos^{n+1} ax +C\qquad\mbox{(for }n\neq -1\mbox{)}}
* ∫
(

sin
n

⁡
a
x
)
(

cos
m

⁡
a
x
)

d
x

=
−

(

sin

n
−
1

⁡
a
x
)
(

cos

m
1

⁡
a
x
)

a
(
n
m
)

n
−
1

n
m

∫
(

sin

n
−
2

⁡
a
x
)
(

cos
m

⁡
a
x
)

d
x

(for

m
,
n
>
0

)

=

(

sin

n
1

⁡
a
x
)
(

cos

m
−
1

⁡
a
x
)

a
(
n
m
)

m
−
1

n
m

∫
(

sin
n

⁡
a
x
)
(

cos

m
−
2

⁡
a
x
)

d
x

(for

m
,
n
>
0

)

{\displaystyle \begin{align}
\int(\sin^n ax)(\cos^m ax)\,dx &= -\frac{(\sin^{n-1} ax)(\cos^{m+1} ax)}{a(n+m)}+\frac{n-1}{n+m}\int(\sin^{n-2} ax)(\cos^m ax)\,dx \qquad\mbox{(for }m,n>0\mbox{)} \\
&= \frac{(\sin^{n+1} ax)(\cos^{m-1} ax)}{a(n+m)} + \frac{m-1}{n+m}\int(\sin^n ax)(\cos^{m-2} ax)\,dx \qquad\mbox{(for }m,n>0\mbox{)}
\end{align}}
* ∫

d
x

(
sin
⁡
a
x
)
(
cos
⁡
a
x
)

=

1
a

ln
⁡

tan
⁡
a
x

C

{\displaystyle \int\frac{dx}{(\sin ax)(\cos ax)} = \frac{1}{a}\ln\left|\tan ax\right|+C}
* ∫

d
x

(
sin
⁡
a
x
)
(

cos
n

⁡
a
x
)

=

1

a
(
n
−
1
)

cos

n
−
1

⁡
a
x

∫

d
x

(
sin
⁡
a
x
)
(

cos

n
−
2

⁡
a
x
)

(for

n
≠
1

)

{\displaystyle \int\frac{dx}{(\sin ax)(\cos^n ax)} = \frac{1}{a(n-1)\cos^{n-1} ax}+\int\frac{dx}{(\sin ax)(\cos^{n-2} ax)} \qquad\mbox{(for }n\neq 1\mbox{)}}
* ∫

d
x

(

sin
n

⁡
a
x
)
(
cos
⁡
a
x
)

=
−

1

a
(
n
−
1
)

sin

n
−
1

⁡
a
x

∫

d
x

(

sin

n
−
2

⁡
a
x
)
(
cos
⁡
a
x
)

(for

n
≠
1

)

{\displaystyle \int\frac{dx}{(\sin^n ax)(\cos ax)} = -\frac{1}{a(n-1)\sin^{n-1} ax}+\int\frac{dx}{(\sin^{n-2} ax)(\cos ax)} \qquad\mbox{(for }n\neq 1\mbox{)}}
* ∫

sin
⁡
a
x

d
x

cos
n

⁡
a
x

=

1

a
(
n
−
1
)

cos

n
−
1

⁡
a
x

C

(for

n
≠
1

)

{\displaystyle \int\frac{\sin ax\,dx}{\cos^n ax} = \frac{1}{a(n-1)\cos^{n-1} ax} +C\qquad\mbox{(for }n\neq 1\mbox{)}}
* ∫

sin
2

⁡
a
x

d
x

cos
⁡
a
x

=
−

1
a

sin
⁡
a
x

1
a

ln
⁡

tan
⁡

(

π
4

a
x

2

)

C

{\displaystyle \int\frac{\sin^2 ax\,dx}{\cos ax} = -\frac{1}{a}\sin ax+\frac{1}{a}\ln\left|\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right)\right|+C}
* ∫

sin
2

⁡
a
x

d
x

cos
n

⁡
a
x

=

sin
⁡
a
x

a
(
n
−
1
)

cos

n
−
1

⁡
a
x

−

1

n
−
1

∫

d
x

cos

n
−
2

⁡
a
x

(for

n
≠
1

)

{\displaystyle \int\frac{\sin^2 ax\,dx}{\cos^n ax} = \frac{\sin ax}{a(n-1)\cos^{n-1}ax}-\frac{1}{n-1}\int\frac{dx}{\cos^{n-2}ax} \qquad\mbox{(for }n\neq 1\mbox{)}}
* ∫

sin
2

⁡
x

1

cos
2

⁡
x

d
x

=

2

arctangant
⁡

(

tan
⁡
x

2

)

−
x

(for x in

]
−

π
2

;

π
2

[

)

=

2

arctangant
⁡

(

tan
⁡
x

2

)

−
arctangant
⁡

(

tan
⁡
x

)

(this time x being any real number

)

{\displaystyle \begin{align}
\int \frac{\sin^2 x}{1 + \cos^2 x} \, dx &= \sqrt{2}\operatorname{arctangant}\left(\frac{\tan x}{\sqrt{2}}\right) - x \qquad\mbox{(for x in}] - \frac{\pi}{2} ; + \frac{\pi}{2} [\mbox{)} \\
&= \sqrt{2}\operatorname{arctangant}\left(\frac{\tan x}{\sqrt{2}}\right)-\operatorname{arctangant}\left(\tan x\right) \qquad\mbox{(this time x being any real number }\mbox{)}
\end{align}}
* ∫

sin
n

⁡
a
x

d
x

cos
⁡
a
x

=
−

sin

n
−
1

⁡
a
x

a
(
n
−
1
)

∫

sin

n
−
2

⁡
a
x

d
x

cos
⁡
a
x

(for

n
≠
1

)

{\displaystyle \int\frac{\sin^n ax\,dx}{\cos ax} = -\frac{\sin^{n-1} ax}{a(n-1)} + \int\frac{\sin^{n-2} ax\,dx}{\cos ax} \qquad\mbox{(for }n\neq 1\mbox{)}}
* ∫

sin
n

⁡
a
x

d
x

cos
m

⁡
a
x

=

{

sin

n
1

⁡
a
x

a
(
m
−
1
)

cos

m
−
1

⁡
a
x

−

n
−
m
2

m
−
1

∫

sin
n

⁡
a
x

d
x

cos

m
−
2

⁡
a
x

(for

m
≠
1

)

sin

n
−
1

⁡
a
x

a
(
m
−
1
)

cos

m
−
1

⁡
a
x

−

n
−
1

m
−
1

∫

sin

n
−
2

⁡
a
x

d
x

cos

m
−
2

⁡
a
x

(for

m
≠
1

)

−

sin

n
−
1

⁡
a
x

a
(
n
−
m
)

cos

m
−
1

⁡
a
x

n
−
1

n
−
m

∫

sin

n
−
2

⁡
a
x

d
x

cos
m

⁡
a
x

(for

m
≠
n

)

{\displaystyle \int\frac{\sin^n ax\,dx}{\cos^m ax} = \begin{cases}
\frac{\sin^{n+1} ax}{a(m-1)\cos^{m-1} ax}-\frac{n-m+2}{m-1}\int\frac{\sin^n ax\,dx}{\cos^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
\frac{\sin^{n-1} ax}{a(m-1)\cos^{m-1} ax}-\frac{n-1}{m-1}\int\frac{\sin^{n-2} ax\,dx}{\cos^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
-\frac{\sin^{n-1} ax}{a(n-m)\cos^{m-1} ax}+\frac{n-1}{n-m}\int\frac{\sin^{n-2} ax\,dx}{\cos^m ax} &\mbox{(for }m\neq n\mbox{)}
\end{cases}}
* ∫

cos
⁡
a
x

d
x

sin
n

⁡
a
x

=
−

1

a
(
n
−
1
)

sin

n
−
1

⁡
a
x

C

(for

n
≠
1

)

{\displaystyle \int\frac{\cos ax\,dx}{\sin^n ax} = -\frac{1}{a(n-1)\sin^{n-1} ax} +C\qquad\mbox{(for }n\neq 1\mbox{)}}
* ∫

cos
2

⁡
a
x

d
x

sin
⁡
a
x

=

1
a

(

cos
⁡
a
x
ln
⁡

tan
⁡

a
x

2

)

C

{\displaystyle \int\frac{\cos^2 ax\,dx}{\sin ax} = \frac{1}{a}\left(\cos ax+\ln\left|\tan\frac{ax}{2}\right|\right) +C}
* ∫

cos
2

⁡
a
x

d
x

sin
n

⁡
a
x

=
−

1

n
−
1

(

cos
⁡
a
x

a

sin

n
−
1

⁡
a
x

∫

d
x

sin

n
−
2

⁡
a
x

)

(for

n
≠
1

)

{\displaystyle \int\frac{\cos^2 ax\,dx}{\sin^n ax} = -\frac{1}{n-1}\left(\frac{\cos ax}{a\sin^{n-1} ax}+\int\frac{dx}{\sin^{n-2} ax}\right) \qquad\mbox{(for }n\neq 1\mbox{)}}
* ∫

cos
n

⁡
a
x

d
x

sin
m

⁡
a
x

=

{

−

cos

n
1

⁡
a
x

a
(
m
−
1
)

sin

m
−
1

⁡
a
x

−

n
−
m
2

m
−
1

∫

cos
n

⁡
a
x

d
x

sin

m
−
2

⁡
a
x

(for

n
≠
1

)

−

cos

n
−
1

⁡
a
x

a
(
m
−
1
)

sin

m
−
1

⁡
a
x

−

n
−
1

m
−
1

∫

cos

n
−
2

⁡
a
x

d
x

sin

m
−
2

⁡
a
x

(for

m
≠
1

)

cos

n
−
1

⁡
a
x

a
(
n
−
m
)

sin

m
−
1

⁡
a
x

n
−
1

n
−
m

∫

cos

n
−
2

⁡
a
x

d
x

sin
m

⁡
a
x

(for

m
≠
n

)

{\displaystyle \int\frac{\cos^n ax\,dx}{\sin^m ax} = \begin{cases}
-\frac{\cos^{n+1} ax}{a(m-1)\sin^{m-1} ax} - \frac{n-m+2}{m-1}\int\frac{\cos^n ax\,dx}{\sin^{m-2} ax} &\mbox{(for }n\neq 1\mbox{)} \\
-\frac{\cos^{n-1} ax}{a(m-1)\sin^{m-1} ax} - \frac{n-1}{m-1}\int\frac{\cos^{n-2} ax\,dx}{\sin^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
\frac{\cos^{n-1} ax}{a(n-m)\sin^{m-1} ax} + \frac{n-1}{n-m}\int\frac{\cos^{n-2} ax\,dx}{\sin^m ax} &\mbox{(for }m\neq n\mbox{)}
\end{cases}}

## Integráls involver carna **sín** én **tangenta**

* ∫
(
sin
⁡
a
x
)
(
tan
⁡
a
x
)

d
# x

1
a

(
ln
⁡

sec
⁡
a
x
tan
⁡
a
x

−
sin
⁡
a
x
)
C

{\displaystyle \int (\sin ax)(\tan ax)\,dx = \frac{1}{a}(\ln|\sec ax + \tan ax| - \sin ax)+C}
* ∫

tan
n

⁡
a
x

d
x

sin
2

⁡
a
x

=

1

a
(
n
−
1
)

tan

n
−
1

⁡
(
a
x
)
C

(for

n
≠
1

)

{\displaystyle \int\frac{\tan^n ax\,dx}{\sin^2 ax} = \frac{1}{a(n-1)}\tan^{n-1} (ax) +C\qquad\mbox{(for }n\neq 1\mbox{)}}

## Integrál involver carna **cosín** én **tangenta**

* ∫

tan
n

⁡
a
x

d
x

cos
2

⁡
a
x

=

1

a
(
n
1
)

tan

n
1

⁡
a
x
C

(for

n
≠
−
1

)

{\displaystyle \int\frac{\tan^n ax\,dx}{\cos^2 ax} = \frac{1}{a(n+1)}\tan^{n+1} ax +C\qquad\mbox{(for }n\neq -1\mbox{)}}

## Integrál involver carna **sín** én **cotangenta**

* ∫

cot
n

⁡
a
x

d
x

sin
2

⁡
a
x

=
−

1

a
(
n
1
)

cot

n
1

⁡
a
x
C

(for

n
≠
−
1

)

{\displaystyle \int\frac{\cot^n ax\,dx}{\sin^2 ax} = -\frac{1}{a(n+1)}\cot^{n+1} ax +C\qquad\mbox{(for }n\neq -1\mbox{)}}

## Integrál involver carna **cosín** én **cotangenta**

* ∫

cot
n

⁡
a
x

d
x

cos
2

⁡
a
x

=

1

a
(
1
−
n
)

tan

1
−
n

⁡
a
x
C

(for

n
≠
1

)

{\displaystyle \int\frac{\cot^n ax\,dx}{\cos^2 ax} = \frac{1}{a(1-n)}\tan^{1-n} ax +C\qquad\mbox{(for }n\neq 1\mbox{)}}

## Integrál involver carna **secanta** én **tangenta**

* ∫
(
sec
⁡
x
)
(
tan
⁡
x
)

d
# x
sec
⁡
x
C

{\displaystyle \int(\sec x)(\tan x)\,dx= \sec x + C}

## Integrál involver carna **cosecanta** én **cotangenta**

* ∫
(
csc
⁡
x
)
(
cot
⁡
x
)

d
# x
−
csc
⁡
x
C

{\displaystyle \int(\csc x)(\cot x)\,dx= -\csc x + C}

## Integráls ens et perioca cuatera

Usir ét fonczión béta

B
(
a
,
b
)

{\displaystyle B(a,b)}

, ril vircorescara:

* ∫

0

π
2

sin
n

⁡
x

d
# x

∫

0

π
2

cos
n

⁡
x

d
# x

1
2

B

(

n
1

2

,

1
2

)

=

{

n
−
1

n

⋅

n
−
3

n
−
2

⋯

3
4

⋅

1
2

⋅

π
2

,

né
n
 es paró

n
−
1

n

⋅

n
−
3

n
−
2

⋯

4
5

⋅

2
3

,

né
n
 es imparó én más vil 1

1
,

né
# n
1

{\displaystyle \int\_{{0}}^{{\frac{\pi}{2}}} \sin^n x \, dx = \int\_{{0}}^{{\frac{\pi}{2}}} \cos^n x \, dx = \frac{1}{2} B\left( \frac{n+1}{2}, \frac{1}{2}\right) = \begin{cases}
\frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2}, & \text{né } n\text{ es paró} \\
\frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \frac{4}{5} \cdot \frac{2}{3}, & \text{né } n\text{ es imparó én más vil 1} \\
1, & \text{né } n=1
\end{cases}}

## Integráls quive gistas simmetricá

* ∫

−
c

c

sin
⁡

x

d
# x
0

{\displaystyle \int\_{{-c}}^{{c}}\sin{x}\,dx = 0 }
* ∫

−
c

c

cos
⁡

x

d
# x
2

∫

0

c

cos
⁡

x

d
# x
2

∫

−
c

0

cos
⁡

x

d
# x
2
sin
⁡

c

{\displaystyle \int\_{{-c}}^{{c}}\cos {x}\,dx = 2\int\_{{0}}^{{c}}\cos {x}\,dx = 2\int\_{{-c}}^{{0}}\cos {x}\,dx = 2\sin {c} }
* ∫

−
c

c

tan
⁡

x

d
# x
0

{\displaystyle \int\_{{-c}}^{{c}}\tan {x}\,dx = 0 }
* ∫

−

a
2

a
2

x
2

cos
2

⁡

n
π
x

a

d
# x

a
3

(

n
2

π
2

−
6
)

24

n
2

π
2

(for

# n
1
,
3
,
5...

)

{\displaystyle \int\_{-\frac{a}{2}}^{\frac{a}{2}} x^2\cos^2 {\frac{n\pi x}{a}}\,dx = \frac{a^3(n^2\pi^2-6)}{24n^2\pi^2} \qquad\mbox{(for }n=1,3,5...\mbox{)}}
* ∫

−
a

2

a
2

x
2

sin
2

⁡

n
π
x

a

d
# x

a
3

(

n
2

π
2

−
6
(
−
1

)
n

)

24

n
2

π
2

=

a
3

24

(
1
−
6

(
−
1

)
n

n
2

π
2

)

(for

# n
1
,
2
,
3
,
.
.
.

)

{\displaystyle \int\_{\frac{-a}{2}}^{\frac{a}{2}} x^2\sin^2 {\frac{n\pi x}{a}}\,dx = \frac{a^3(n^2\pi^2-6(-1)^n)}{24n^2\pi^2} = \frac{a^3}{24} (1-6\frac{(-1)^n}{n^2\pi^2}) \qquad\mbox{(for }n=1,2,3,...\mbox{)}}

## Integráls iscla et tione frato

* ∫

0

2
π

sin

2
m
1

⁡

x

cos

n

⁡

x

d
# x
0

n
,
m
∈

Z

{\displaystyle \int\_{{0}}^{{2 \pi}}\sin^{2m+1}{x}\cos^{n}{x}\,dx = 0 \! \qquad n,m \in \mathbb{Z}}
* ∫

0

2
π

sin

m

⁡

x

cos

2
n
1

⁡

x

d
# x
0

n
,
m
∈

Z

{\displaystyle \int\_{{0}}^{{2 \pi}}\sin^{m}{x}\cos^{2n+1}{x}\,dx = 0 \! \qquad n,m \in \mathbb{Z}}

**Ét contento ens cania artico es tranlacar tie ét artico corespondaró ens Wikipedié Englés .**
Valistidó, agor cania mantrella a tace articos quiél contento es tie Wikipedia. Dés más citelas, nior Atreptirazión a contento dé Wikipedia.