avewright/proofwiki-math huggingface dataset:
Theorem: Primitive of Reciprocal of x by Power of x plus Power of a

:$\ds \int \frac {\d x} {x \paren {x^n + a^n} } = \frac 1 {n a^n} \ln \size {\frac {x^n} {x^n + a^n} } + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^n + a^n} }
      | r = \int \frac {a^n \rd x} {a^n x \paren {x^2 + a^2} }
      | c = multiplying top and bottom by $a^n$
}}
{{eqn | r = \int \frac {\paren {x^n + a^n - x^n} \rd x} {a^n x \paren {x^n + a^n} }
      | c = adding and subtracting $x^n$
}}
{{eqn | r = \frac 1 {a^n} \int \frac {\paren {x^n + a^n} \rd x} {x \paren {x^n + a^n} } - \frac 1 {a^n} \int \frac {x^n \rd x} {x \paren {x^n + a^n} }
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac 1 {a^n} \int \frac {\d x} x - \frac 1 {a^n} \int \frac {x^{n - 1} \rd x} {x^n + a^n}
      | c = simplifying
}}
{{eqn | r = \frac 1 {a^n} \ln \size x - \frac 1 {a^n} \int \frac {x^{n - 1} \rd x} {x^n + a^n} + C
      | c = Primitive of Reciprocal
}}
{{eqn | r = \frac 1 {a^n} \ln \size x - \frac 1 {a^n} \paren {\frac 1 n \ln \size {x^n + a^n} } + C
      | c = Primitive of $\dfrac {x^{n - 1}} {x^n + a^n}$
}}
{{eqn | r = \frac 1 {n a^n} \ln \size {x^n} - \frac 1 {n a^n} \ln \size {x^n + a^n} + C
      | c = Logarithm of Power
}}
{{eqn | r = \frac 1 {n a^n} \ln \paren {\frac {x^n} {x^n + a^n} } + C
      | c = Difference of Logarithms
}}
{{end-eqn}}
{{qed}}
Theorem: Absolute Value of Even Power

Let $x \in \R$ be a real number.
Let $n \in \Z$ be an integer.
Then:
:$\size {x^{2 n} } = x^{2 n}$

Proof:
From Even Power is Non-Negative:
:$x^{2 n} \ge 0$
The result follows from the definition of absolute value.
{{qed}}
Category:Absolute Value Function
4q0q24ahaqcyqiyl1er3fltsy5q6dxm
Theorem: Primitive of Power of x less one over Power of x plus Power of a

:$\ds \int \frac {x^{n - 1} \rd x} {x^n + a^n} = \frac 1 n \ln \size {x^n + a^n} + C$

Proof:
{{begin-eqn}}
{{eqn | l = u
      | r = x^n + a^n
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = n x^{n - 1} 
      | c = Power Rule for Derivatives and Derivative of Constant
}}
{{eqn | ll= \leadsto
      | l = \int \frac {x^{n - 1} \rd x} {x^n + a^n}
      | r = \frac 1 n \ln \size {x^n + a^n} + C
      | c = Primitive of Function under its Derivative
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Power of x over Power of Power of x plus Power of a

:$\ds \int \frac {x^m \rd x} {\paren {x^n + a^n}^r} = \int \frac {x^{m - n} \rd x} {\paren {x^n + a^n}^{r - 1} } - a^n \int \frac {x^{m - n} \rd x} {\paren {x^n + a^n}^r}$

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {\paren {x^n + a^n}^r}
      | r = \int \frac {x^{m - n} x^n \rd x} {\paren {x^n + a^n}^r}
      | c = 
}}
{{eqn | r = \int \frac {x^{m - n} \paren {x^n + a^n - a^n} \rd x} {\paren {x^n + a^n}^r}
      | c = 
}}
{{eqn | r = \int \frac {x^{m - n} \paren {x^n + a^n} \rd x} {\paren {x^n + a^n}^r} - a_n \int \frac {x^{m - n} \rd x} {\paren {x^n + a^n}^r}
      | c = Linear Combination of Primitives
}}
{{eqn | r = \int \frac {x^{m - n} \rd x} {\paren {x^n + a^n}^{r - 1} } - a^n \int \frac {x^{m - n} \rd x} {\paren {x^n + a^n}^r}
      | c = simplification
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Power of x by Power of Power of x plus Power of a

:$\ds \int \frac {\d x} {x^m \paren {x^n + a^n}^r} = \frac 1 {a^n} \int \frac {\d x} {x^m \paren {x^n + a^n}^{r - 1} } - \frac 1 {a^n} \int \frac {\d x} {x^{m - n} \paren {x^n + a^n}^r}$

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \paren {x^n + a^n}^r}
      | r = \int \frac {a^n \rd x} {a^n x^m \paren {x^n + a^n}^r}
      | c = multiplying top and bottom by $a^n$
}}
{{eqn | r = \int \frac {\paren {x^n + a^n - x^n} \rd x} {a^n x^m \paren {x^n + a^n}^r}
      | c = adding and subtracting $x^n$
}}
{{eqn | r = \frac 1 {a^n} \int \frac {\paren {x^n + a^n} \rd x} {x^m \paren {x^n + a^n}^r} - \frac 1 {a^n} \int \frac {x^n \rd x} {x^m \paren {x^n + a^n}^r}
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac 1 {a^n} \int \frac {\d x} {x^m \paren {x^n + a^n}^{r - 1} } - \frac 1 {a^n} \int \frac {\d x} {x^{m - n} \paren {x^n + a^n}^r}
      | c = simplification
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of x by Root of Power of x plus Power of a

:$\ds \int \frac {\d x} {x \sqrt {x^n + a^n} } = \frac 1 {n \sqrt {a^n} } \ln \size {\frac {\sqrt {x^n + a^n} - \sqrt {a^n} } {\sqrt {x^n + a^n} + \sqrt {a^n} } } + C$

Proof:
{{begin-eqn}}
{{eqn | l = u
      | r = \sqrt {x^n + a^n}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = \frac {n x^{n - 1} } {2 \sqrt {x^n + a^n} } 
      | c = Derivative of Power, Chain Rule for Derivatives
}}
{{eqn | ll= \leadsto
      | l = \int \frac {\d x} {x \sqrt {x^n + a^n} }
      | r = \int \frac {2 \sqrt {x^n + a^n} \rd u} {n x^{n - 1} x \sqrt {x^n + a^n} }
      | c = Integration by Substitution
}}
{{eqn | r = \int \frac {2 \rd u} {n \paren {u^2 - a^n} }
      | c = completing substitution and simplifying
}}
{{eqn | r = \frac 2 n \int \frac {\d u} {\paren {u^2 - \paren {\sqrt {a^n} } ^2} }
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac 2 n \paren {\frac 1 {2 \sqrt {a^n} } \ln \size {\frac {u - \sqrt {a^n} } {u + \sqrt {a^n} } } } + C
      | c = Primitive of $\dfrac 1 {x^2 - a^2}$
}}
{{eqn | r = \frac 1 {n \sqrt {a^n} } \ln \size {\frac {\sqrt {x^n + a^n} - \sqrt {a^n} } {\sqrt {x^n + a^n} + \sqrt {a^n} } } + C
      | c = substituting for $u$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of x by Power of x minus Power of a

:$\ds \int \frac {\d x} {x \paren {x^n - a^n} } = \frac 1 {n a^n} \ln \size {\frac {x^n - a^n} {x^n} } + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {x^n - a^n} }
      | r = \int \frac {a^n \rd x} {a^n x \paren {x^2 - a^2} }
      | c = multiplying top and bottom by $a^n$
}}
{{eqn | r = \int \frac {\paren {-\paren {x^n - a^n} + x^n} \rd x} {a^n x \paren {x^n - a^n} }
      | c = adding and subtracting $x^n$
}}
{{eqn | r = \frac {-1} {a^n} \int \frac {\paren {x^n - a^n} \rd x} {x \paren {x^n - a^n} } + \frac 1 {a^n} \int \frac {x^n \rd x} {x \paren {x^n - a^n} }
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac {-1} {a^n} \int \frac {\d x} x + \frac 1 {a^n} \int \frac {x^{n - 1} \rd x} {x^n - a^n}
      | c = simplifying
}}
{{eqn | r = \frac {-1} {a^n} \ln \size x + \frac 1 {a^n} \int \frac {x^{n - 1} \rd x} {x^n - a^n} + C
      | c = Primitive of Reciprocal
}}
{{eqn | r = \frac {-1} {a^n} \ln \size x + \frac 1 {a^n} \paren {\frac 1 n \ln \size {x^n - a^n} } + C
      | c = Primitive of $\dfrac {x^{n - 1} } {x^n - a^n}$
}}
{{eqn | r = \frac {-1} {n a^n} \ln \size {x^n} + \frac 1 {n a^n} \ln \size {x^n - a^n} + C
      | c = Logarithm of Power
}}
{{eqn | r = \frac 1 {n a^n} \ln \size {\frac {x^n - a^n} {x^n} } + C
      | c = Difference of Logarithms
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Power of x less one over Power of x minus Power of a

:$\ds \int \frac {x^{n - 1} \rd x} {x^n - a^n} = \frac 1 n \ln \size {x^n - a^n} + C$

Proof:
{{begin-eqn}}
{{eqn | l = u
      | r = x^n - a^n
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = n x^{n - 1} 
      | c = Power Rule for Derivatives and Derivative of Constant
}}
{{eqn | ll= \leadsto
      | l = \int \frac {x^{n - 1} \rd x} {x^n - a^n}
      | r = \frac 1 n \ln \size {x^n - a^n} + C
      | c = Primitive of Function under its Derivative
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Power of x over Power of Power of x minus Power of a

:$\ds \int \frac {x^m \rd x} {\paren {x^n - a^n}^r} = a^n \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^r} + \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^{r - 1} }$

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {x^m \rd x} {\paren {x^n - a^n}^r}
      | r = \int \frac {x^{m - n} x^n \rd x} {\paren {x^n + a^n}^r}
      | c = 
}}
{{eqn | r = \int \frac {x^{m - n} \paren {x^n - a^n + a^n} \rd x} {\paren {x^n - a^n}^r}
      | c = 
}}
{{eqn | r = \int \frac {x^{m - n} \paren {x^n - a^n} \rd x} {\paren {x^n - a^n}^r} + a^n \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^r}
      | c = Linear Combination of Primitives
}}
{{eqn | r = a^n \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^r} + \int \frac {x^{m - n} \rd x} {\paren {x^n - a^n}^{r - 1} }
      | c = simplification
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Power of x by Power of Power of x minus Power of a

:$\ds \int \frac {\d x} {x^m \ \paren {x^n - a^n}^r} = \frac 1 {a^n} \int \frac {\d x} {x^{m - n} \paren {x^n - a^n}^r} - \frac 1 {a^n} \int \frac {\d x} {x^m \paren {x^n - a^n}^{r - 1} }$

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^m \ \paren {x^n - a^n}^r}
      | r = \int \frac {a^n \rd x} {a^n x^m \ \paren {x^n - a^n}^r}
      | c = multiplying top and bottom by $a^n$
}}
{{eqn | r = \int \frac {\paren {x^n - \paren {x^n - a^n} } \rd x} {a^n x^m \ \paren {x^n + a^n}^r}
      | c = adding and subtracting $x^n$
}}
{{eqn | r = \frac 1 {a^n} \int \frac {x^n \rd x} {x^m \ \paren {x^n - a^n}^r} - \frac 1 {a^n} \int \frac {\paren {x^n - a^n} \rd x} {x^m \ \paren {x^n - a^n}^r}
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac 1 {a^n} \int \frac {\d x} {x^{m - n} \paren {x^n - a^n}^r} - \frac 1 {a^n} \int \frac {\d x} {x^m \paren {x^n - a^n}^{r - 1} }
      | c = simplification
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of x by Root of Power of x minus Power of a

:$\ds \int \frac {\d x} {x \sqrt {x^n - a^n} } = \frac 2 {n \sqrt {a^n} } \arccos \sqrt {\frac {a^n} {x^n} }$

Proof:
{{begin-eqn}}
{{eqn | l = u
      | r = \sqrt {x^n - a^n}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = \frac {n x^{n - 1} } {2 \sqrt {x^n - a^n} } 
      | c = Derivative of Power, Chain Rule for Derivatives
}}
{{eqn | ll= \leadsto
      | l = \int \frac {\d x} {x \sqrt {x^n - a^n} }
      | r = \int \frac {2 \sqrt {x^n - a^n} \rd u} {n x^{n - 1} x \sqrt {x^n - a^n} }
      | c = Integration by Substitution
}}
{{eqn | r = \int \frac {2 \rd u} {n \paren {u^2 + a^n} }
      | c = completing substitution and simplifying
}}
{{eqn | r = \frac 2 n \int \frac {\d u} {\paren {u^2 + \paren {\sqrt {a^n} }^2} }
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac 2 n \paren {\frac 1 {\sqrt {a^n} } \arctan \frac u {\sqrt {a^n} } } + C
      | c = Primitive of $\dfrac 1 {x^2 + a^2}$
}}
{{eqn | n = 1
      | r = \frac 2 {n \sqrt {a^n} } \arctan \frac {\sqrt {x^n - a^n} } {\sqrt {a^n} } + C
      | c = substituting for $u$
}}
{{end-eqn}}
Now:
{{begin-eqn}}
{{eqn | l = y
      | r = \arctan \frac {\sqrt {x^n - a^n} } {\sqrt {a^n} }
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \tan y
      | r = \frac {\sqrt {x^n - a^n} } {\sqrt {a^n} }
      | c = {{Defof|Real Arctangent}}
}}
{{eqn | ll= \leadsto
      | l = \tan^2 y
      | r = \frac {x^n - a^n} {a^n}
      | c = squaring both sides
}}
{{eqn | ll= \leadsto
      | l = \tan^2 y
      | r = \frac {x^n} {a^n} - 1
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 1 + \tan^2 y
      | r = \frac {x^n} {a^n}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \sec^2 y
      | r = \frac {x^n} {a^n}
      | c = Difference of Squares of Secant and Tangent
}}
{{eqn | ll= \leadsto
      | l = \cos^2 y
      | r = \frac {a^n} {x^n}
      | c = {{Defof|Secant Function}}
}}
{{eqn | ll= \leadsto
      | l = \cos y
      | r = \frac {\sqrt {a^n} } {\sqrt {x^n} }
      | c = 
}}
{{eqn | ll= \leadsto
      | l = y
      | r = \arccos \frac {\sqrt {a^n} } {\sqrt {x^n} }
      | c = {{Defof|Real Arccosine}}
}}
{{eqn | ll= \leadsto
      | l = \int \frac {\d x} {x \sqrt {x^n - a^n} }
      | r = \frac 2 {n \sqrt {a^n} } \arccos \sqrt {\frac {a^n} {x^n} }
      | c = substituting in $(1)$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Power of x over Even Power of x minus Even Power of a

{{begin-eqn}}
{{eqn | l = \int \frac {x^{p - 1} \rd x} {x^{2 m} - a^{2 m} }
      | r = \frac 1 {2 m a^{2 m - p} } \sum_{k \mathop = 1}^{m - 1} \map \cos {\frac {k p \pi} m} \map \ln {x^2 - 2 a x \map \cos {\frac {k \pi} m} + a^2}
}}
{{eqn | o =
      | ro= -
      | r = \frac 1 {m a^{2 m - p} } \sum_{k \mathop = 1}^{m - 1} \map \sin {\frac {k p \pi} m} \map \arctan {\frac {x - a \map \cos {\dfrac {k \pi} m} } {a \map \sin {\dfrac {k \pi} m} } }
      | c = 
}}
{{eqn | o =
      | ro= +
      | r = \frac 1 {2 m a^{2 m - p} } \paren {\map \ln {x - a} + \paren {-1}^p \map \ln {x + a} }
      | c = 
}}
{{end-eqn}}
where $0 < p \le 2 m$.

Proof:
The integrand is a rational function.
It has simple poles at $x = \omega_k a$ where $\omega_k = e^{\pi i k /m} $, $k = 0, 1, \ldots, 2 m - 1$ are the $2m$'th roots of unity:
{{begin-eqn}}
{{eqn | l = \map f x
      | r = \dfrac {x^{p - 1} } {x^{2 m} - a^{2 m} }
      | c = 
}}
{{eqn | r = \dfrac {x^{p - 1} } {\ds \prod_{k \mathop = 0}^{2 m - 1} \paren {x - \omega_k a} }
      | c = 
}}
{{end-eqn}}
Its residue at $x = \omega_k$ is then:
:$\Res f {\omega_k a} = \dfrac {\paren {\omega_k a}^{p - 2m} } {2 m}$
so that:
:$\ds \map f x = \sum_{k \mathop = 0}^{2 m - 1} \dfrac {\paren {\omega_k a}^{p - 2m} } {2 m \paren {x - \omega_k a} }$
which has a primitive:
:$\ds \map F x = \sum_{k \mathop = 0}^{2 m - 1} \dfrac {\paren {\omega_k a}^{p - 2m} } {2 m} \map \ln {x - \omega_k a}$
For positive real $x, a \in \R_{>0}$ we can write:
:$\map \ln {x - \omega_k a} = \dfrac 1 2 \map \ln {x^2 - 2 a x \map \cos {\dfrac {2 \pi k} m} + a^2} - i \map \arctan {\dfrac {a \map \sin {\dfrac {2 \pi k} m} } {x - a \map \cos {\dfrac {2 \pi k} m} } }$
:$\paren {\omega_k a}^{p - 2 m} = a^{p - 2 m} \paren {\map \cos {\dfrac {p k \pi} m} + i \map \sin {\dfrac {p k \pi} m} }$
and the formula follows.
{{finish|Fill in the details}}
Theorem: Primitive of x by Sine of a x

:$\ds \int x \sin a x \rd x = \frac {\sin a x} {a^2} - \frac {x \cos a x} a + C$

Proof:
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
      | r = x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = 1
      | c = Derivative of Identity Function
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = \sin a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = -\frac {\cos a x} a
      | c = Primitive of $\sin a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \map \sin {a x} \rd x
      | r = x \paren {-\frac {\cos a x} a} - \int \paren {-\frac {\cos a x} a} \times 1 \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = -\frac {x \cos a x} a + \frac 1 a \int \cos a x \rd x + C
      | c = Linear Combination of Primitives
}}
{{eqn | r = -\frac {x \cos a x} a + \frac 1 a \paren {\frac {\sin a x} a} + C
      | c = Primitive of $\cos a x$
}}
{{eqn | r = \frac {\sin a x} {a^2} - \frac {x \cos a x} a + C
      | c = simplification
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x squared by Sine of a x

:$\ds \int x^2 \sin a x \rd x = \frac {2 x \sin a x} {a^2} + \paren {\frac 2 {a^3} - \frac {x^2} a} \cos a x + C$

Proof:
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
      | r = x^2
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = 2 x
      | c = Derivative of Power
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = \sin a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = -\frac {\cos a x} a
      | c = Primitive of $\sin a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^2 \sin a x \rd x
      | r = x^2 \paren {-\frac {\cos a x} a} - \int 2 x \paren {-\frac {\cos a x} a} \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = -\frac {x^2} a \cos a x + \frac 2 a \int x \cos a x \rd x + C
      | c = Linear Combination of Primitives
}}
{{eqn | r = -\frac {x^2} a \cos a x + \frac 2 a \paren {\frac {\cos a x} {a^2} + \frac {x \sin a x} a} + C
      | c = Primitive of $x \cos a x$
}}
{{eqn | r = \frac {2 x \sin a x} {a^2} + \paren {\frac 2 {a^3} - \frac {x^2} a} \cos a x + C
      | c = simplification
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x cubed by Sine of a x

:$\ds \int x^3 \sin a x \rd x = \paren {\frac {3 x^2} {a^2} - \frac 6 {a^4} } \sin a x + \paren {\frac {6 x} {a^3} - \frac {x^3} a} \cos a x + C$

Proof:
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
      | r = x^3
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = 3 x^2
      | c = Derivative of Power
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = \sin a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = -\frac {\cos a x} a
      | c = Primitive of $\sin a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^3 \sin a x \rd x
      | r = x^3 \paren {-\frac {\cos a x} a} - \int 3 x^2 \paren {-\frac {\cos a x} a} \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = -\frac {x^3} a \cos a x + \frac 3 a \int x^2 \cos a x \rd x + C
      | c = Linear Combination of Primitives
}}
{{eqn | r = -\frac {x^3} a \cos a x + \frac 3 a \paren {\frac {2 x} {a^2} \cos a x + \paren {\frac {x^2} a - \frac 2 {a^3} } \sin a x} + C
      | c = Primitive of $x^2 \cos a x$
}}
{{eqn | r = \paren {\frac {3 x^2} {a^2} - \frac 6 {a^4} } \sin a x + \paren {\frac {6 x} {a^3} - \frac {x^3} a} \cos a x + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Sine of a x over x squared

:$\ds \int \frac {\sin a x \rd x} {x^2} = -\frac {\sin a x} x + a \int \frac {\cos a x \rd x} x + C$

Proof:
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
      | r = \sin a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = a \cos a x
      | c = Derivative of $\sin a x$
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = \frac 1 {x^2}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = - \frac 1 x
      | c = Primitive of Power
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\sin a x \rd x} {x^2}
      | r = \int \sin a x \frac 1 {x^2} \rd x
      | c = 
}}
{{eqn | r = \sin a x \paren {-\frac 1 x} - \int \paren {-\frac 1 x} a \cos a x \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = -\frac {\sin a x} x + a \int \frac {\cos a x \rd x} x + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Sine of a x/Logarithm of Cosecant minus Cotangent Form

:$\ds \int \frac {\d x} {\sin a x} = \frac 1 a \ln \size {\csc a x - \cot a x} + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin x}
      | r = \int \csc x \rd x
      | c = {{Defof|Real Cosecant Function}}
}}
{{eqn | r = \ln \size {\csc x - \cot x} + C
      | c = Primitive of $\csc x$: Cosecant minus Cotangent Form
}}
{{eqn | ll= \leadsto
      | l = \int \frac {\d x} {\sin a x}
      | r = \frac 1 a \ln \size {\csc a x - \cot a x} + C
      | c = Primitive of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Sine of a x/Logarithm of Tangent Form

:$\ds \int \frac {\d x} {\sin a x} = \frac 1 a \ln \size {\tan \frac {a x} 2} + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sin x}
      | r = \int \csc x \rd x
      | c = {{Defof|Real Cosecant Function}}
}}
{{eqn | r = \ln \size {\tan \frac x 2} + C
      | c = Primitive of $\csc x$: Tangent Form
}}
{{eqn | ll= \leadsto
      | l = \int \frac {\d x} {\sin a x}
      | r = \frac 1 a \ln \size {\tan \frac {a x} 2} + C
      | c = Primitive of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x over Sine of a x

{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\sin a x}
      | r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C
      | c = 
}}
{{eqn | r = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} {18} + \frac {7 \paren {a x}^5} {1800} \cdots} + C
      | c = 
}}
{{end-eqn}}

Proof:
{{begin-eqn}}
{{eqn | l = \csc x
      | r = \sum_{n \mathop = 0}^\infty \dfrac {B_{2 n} \paren {-1}^{n - 1} x^{2 n - 1} 2 \paren {2^{2 n - 1} - 1} } {\paren {2 n}!}
      | c = Power Series Expansion for Cosecant Function
}}
{{eqn | ll= \leadsto
      | l = \dfrac x {\sin a x}
      | r = x \sum_{n \mathop = 0}^\infty \dfrac {B_{2 n} \paren {-1}^{n - 1} \paren {a x}^{2 n - 1} 2 \paren {2^{2 n - 1} - 1} } {\paren {2 n}!}
      | c = Cosecant is Reciprocal of Sine
}}
{{eqn | r = \dfrac 1 a \sum_{n \mathop = 0}^\infty \dfrac {B_{2 n} \paren {-1}^{n - 1} \paren {a x}^{2 n} 2 \paren {2^{2 n - 1} - 1} } {\paren {2 n}!}
}}
{{eqn | ll= \leadsto
      | l = \int \dfrac x {\sin a x} \rd x
      | r = \dfrac 1 {a^2} \sum_{n \mathop = 0}^\infty \dfrac {B_{2 n} \paren {-1}^{n - 1} \paren {a x}^{2 n + 1} 2 \paren {2^{2 n - 1} - 1} } {\paren {2 n + 1}!}
      | c = Primitive of Power, Linear Combination of Primitives
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Square of Sine of a x

:$\ds \int \sin^2 a x \rd x = \frac x 2 - \frac {\sin 2 a x} {4 a} + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \sin^2 x \rd x
      | r = \frac x 2 - \frac {\sin 2 x} 4 + C
      | c = Primitive of $\sin^2 x$
}}
{{eqn | ll= \leadsto
      | l = \int \sin^2 a x \rd x
      | r = \frac 1 a \paren {\frac {a x} 2 - \frac {\sin 2 a x} 4} + C
      | c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac x 2 - \frac {\sin 2 a x} {4 a} + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x by Square of Sine of a x

:$\ds \int x \sin^2 a x \rd x = \frac {x^2} 4 - \frac {x \sin 2 a x} {4 a} - \frac {\cos 2 a x} {8 a^2} + C$

Proof:
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
      | r = x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = 1
      | c = Derivative of Identity Function
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = \sin^2 a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = \frac x 2 - \frac {\sin 2 a x} {4 a}
      | c = Primitive of $\sin^2 a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \sin^2 a x \rd x
      | r = x \paren {\frac x 2 - \frac {\sin 2 a x} {4 a} } - \int \paren {\frac x 2 - \frac {\sin 2 a x} {4 a} } \times 1 \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = \frac {x^2} 2 - \frac {x \sin 2 a x} {4 a} - \frac 1 2 \int x \rd x + \frac 1 {4 a} \int \sin 2 a x \rd x + C
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac {x^2} 2 - \frac {x \sin 2 a x} {4 a} - \frac {x^2} 4 + \frac 1 {4 a} \int \sin 2 a x \rd x + C
      | c = Primitive of Power
}}
{{eqn | r = \frac {x^2} 2 - \frac {x \sin 2 a x} {4 a} - \frac {x^2} 4 + \frac 1 {4 a} \frac {\paren {-\cos 2 a x} } {2 a} + C
      | c = Primitive of $\sin a x$
}}
{{eqn | r = \frac {x^2} 4 - \frac {x \sin 2 a x} {4 a} - \frac {\cos 2 a x} {8 a^2} + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Fourth Power of Sine of a x

:$\ds \int \sin^4 a x \rd x = \frac {3 x} 8 - \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \sin^4 a x \rd x
      | r = \int \paren {\frac {3 - 4 \cos 2 a x + \cos 4 a x} 8} \rd x
      | c = Power Reduction Formula for $\sin^4$
}}
{{eqn | r = \frac 3 8 \int \rd x - \frac 1 2 \int \cos 2 a x \rd x + \frac 1 8 \int \cos 4 a x \rd x
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac {3 x} 8 - \frac 1 2 \int \cos 2 a x \rd x + \frac 1 8 \int \cos 4 a x \rd x + C
      | c = Primitive of Constant
}}
{{eqn | r = \frac {3 x} 8 - \frac 1 2 \paren {\frac {\sin 2 a x} {2 a} } + \frac 1 8 \paren {\frac {\sin 4 a x} {4 a} } + C
      | c = Primitive of $\cos a x$
}}
{{eqn | r = \frac {3 x} 8 - \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}