hcju/proofwikips huggingface dataset:
Primitive of Exponential of a x by Power of Sine of b x
:$\displaystyle \int e^{a x} \sin^n b x \ \mathrm d x = \frac {e^{a x} \sin^{n - 1} b x} {a^2 + n^2 b^2} \left({a \sin b x - n b \cos b x}\right) + \frac {n \left({n - 1}\right) b^2} {a^2 + n^2 b^2} \int e^{a x} \sin^{n - 2} b x \ \mathrm d x + C$
Primitive of Exponential of a x by Power of Cosine of b x
:$\displaystyle \int e^{a x} \cos^n b x \rd x = \frac {e^{a x} \cos^{n - 1} b x} {a^2 + n^2 b^2} \paren {a \cos b x + n b \sin b x} + \frac {n \paren {n - 1} b^2} {a^2 + n^2 b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C$
Primitive of Logarithm of x
:$\ds \int \ln x \rd x = x \ln x - x + C$
Primitive of x by Logarithm of x
:$\displaystyle \int x \ln x \rd x = \frac {x^2} 2 \left({\ln x - \frac 1 2}\right) + C$
Primitive of Power of x by Logarithm of x
:$\displaystyle \int x^m \ln x \rd x = \frac {x^{m + 1} } {m + 1} \left({\ln x - \frac 1 {m + 1} }\right) + C$ where $m \ne -1$.
Primitive of Logarithm of x over x
:$\displaystyle \int \frac {\ln x} x \ \mathrm d x = \frac {\ln^2 x} 2 + C$
Primitive of Square of Logarithm of x
:$\displaystyle \int \ln^2 x \ \mathrm d x = x \ln^2 x - 2 x \ln x + 2 x + C$
Primitive of Power of Logarithm of x over x
:$\displaystyle \int \frac {\ln^n x} x \ \mathrm d x = \frac {\ln^{n + 1} x} {n + 1} + C$
Primitive of Reciprocal of x by Logarithm of x
:$\displaystyle \int \frac {\mathrm d x} {x \ln x} = \ln \left({\ln x}\right) + C$
Primitive of Power of x over Logarithm of x
:$\displaystyle \int \frac {x^m \rd x} {\ln x} = \map \ln {\ln x} + \paren {m + 1} \ln x + \sum_{k \mathop \ge 2}^n \frac {\paren {m + 1}^k \paren {\ln x}^k} {k \times k!} + C$
Primitive of Logarithm of x squared plus a squared
:$\displaystyle \int \map \ln {x^2 + a^2} \rd x = x \map \ln {x^2 + a^2} - 2 x + 2 a \arctan \frac x a + C$
Primitive of Power of x by Logarithm of x squared plus a squared
:$\displaystyle \int x^m \map \ln {x^2 + a^2} \rd x = \frac {x^{m + 1} \map \ln {x^2 + a^2} } {m + 1} - \frac 2 {m + 1} \int \frac {x^{m + 2} } {x^2 + a^2} \rd x$
Primitive of Power of x by Logarithm of x squared minus a squared
:$\displaystyle \int x^m \, \map \ln {x^2 - a^2} \rd x = \frac {x^{m + 1} \, \map \ln {x^2 - a^2} } {m + 1} -  \frac 2 {m + 1} \int \frac {x^{m + 2} } {x^2 - a^2} \rd x$
Primitive of Hyperbolic Sine of a x
:$\displaystyle \int \sinh a x \ \mathrm d x = \frac {\cosh a x} a + C$
Primitive of Hyperbolic Cosine of a x
:$\displaystyle \int \cosh a x \ \mathrm d x = \frac {\sinh a x} a + C$
Primitive of Hyperbolic Tangent of a x
:$\displaystyle \int \tanh a x \rd x = \frac {\ln \size {\cosh a x} } a + C$
Primitive of Hyperbolic Cotangent of a x
:$\displaystyle \int \coth a x \ \mathrm d x = \frac {\ln \left\vert{\sinh a x}\right\vert} a + C$
Primitive of Hyperbolic Cosecant of a x
:$\displaystyle \int \csch a x \rd x = \frac 1 a \ln \size {\tanh \frac {a x} 2} + C$
Primitive of x by Hyperbolic Sine of a x
:$\displaystyle \int x \sinh a x \ \mathrm d x = \frac {x \cosh a x} a - \frac {\sinh a x} {a^2} + C$
Primitive of x by Hyperbolic Cosine of a x
:$\displaystyle \int x \cosh a x \rd x = \frac {x \sinh a x} a - \frac {\cosh a x} {a^2} + C$
Primitive of x squared by Hyperbolic Cosine of a x
:$\displaystyle \int x^2 \cosh a x \ \mathrm d x = \left({\frac {x^2} a + \frac 2 {a^3} }\right) \sinh a x - \frac {2 x \cosh a x} {a^2} + C$
Primitive of Hyperbolic Cosine of a x over x
{{begin-eqn}} {{eqn | l = \int \frac {\cosh a x \ \mathrm d x} x       | r = \ln \left\vert{x}\right\vert + \sum_{k \mathop \ge 1} \frac { \left({a x}\right)^{2 k} } {\left({2 k}\right) \left({2 k}\right)!} + C }} {{eqn | r = \ln \left\vert{x}\right\vert + \frac {\left({a x}\right)^2} {2 \times 2!} + \frac {\left({a x}\right)^4} {4 \times 4!} + \frac {\left({a x}\right)^6} {6 \times 6!} + \cdots + C }} {{end-eqn}}
Primitive of Hyperbolic Tangent of a x over x
{{begin-eqn}} {{eqn | l = \int \frac {\tanh a x \ \mathrm d x} x       | r = \sum_{k \mathop \ge 1} \frac {\left({-1}\right)^{k - 1} 2^{2 k} \left({2^{2 k} - 1}\right) B_k \left({a x}\right)^{2 k - 1} } {\left({2 k - 1}\right) \left({2 k}\right)!} + C }} {{eqn | r = a x - \frac {\left({a x}\right)^3} 9 + \frac {2 \left({a x}\right)^5} {75} - \cdots + C }} {{end-eqn}} where $B_k$ denotes the $k$th Bernoulli number.
Primitive of Hyperbolic Cotangent of a x over x
{{begin-eqn}} {{eqn | l = \int \frac {\coth a x \ \mathrm d x} x       | r = \sum_{k \mathop \ge 0} \frac {\left({-1}\right)^k 2^{2 k} B_k \left({a x}\right)^{2 k - 1} } {\left({2 k - 1}\right) \left({2 k}\right)!} + C }} {{eqn | r = -\frac 1 {a x} + \frac {a x} 3 - \frac {\left({a x}\right)^3} {135} + \cdots + C }} {{end-eqn}} where $B_k$ denotes the $k$th Bernoulli number.
Primitive of Hyperbolic Secant of a x over x
{{begin-eqn}} {{eqn | l = \int \frac {\operatorname{sech} a x \ \mathrm d x} x       | r = \ln \left\vert{x}\right\vert + \sum_{k \mathop \ge 1} \frac {\left({-1}\right)^k E_k \left({a x}\right)^{2 k} } {\left({2 k}\right) \left({2 k}\right)!} + C }} {{eqn | r = \ln \left\vert{x}\right\vert - \frac {\left({a x}\right)^2} 4 + \frac {\left({a x}\right)^4} {96} - \frac {\left({a x}\right)^6} {4320} + \cdots + C }} {{end-eqn}} where $E_k$ denotes the $k$th Euler number.
Primitive of Hyperbolic Cosecant of a x over x
{{begin-eqn}} {{eqn | l = \int \frac {\operatorname{csch} a x \ \mathrm d x} x       | r = \sum_{k \mathop \ge 0} \frac {\left({-1}\right)^k 2 \left({2^{2 k - 1} - 1}\right) B_k \left({a x}\right)^{2 k - 1} } {\left({2 k - 1}\right) \left({2 k}\right)!} + C }} {{eqn | r = - \frac 1 {a x} - \frac {a x} 6 + \frac {7 \left({a x}\right)^3} {1080} + \cdots + C }} {{end-eqn}} where $E_k$ denotes the $k$th Bernoulli number.
Derivative of Hyperbolic Sine of a x
:$\map {D_x} {\sinh a x} = a \cosh a x$
Derivative of Hyperbolic Cosine of a x
:$\map {\dfrac \d {\d x} } {\cosh a x} = a \sinh a x$
Primitive of Hyperbolic Sine of a x over x squared
:$\displaystyle \int \frac {\sinh a x \ \mathrm d x} {x^2} = -\frac {\sinh a x} x + a \int \frac {\cosh a x \ \mathrm d x} x$
Primitive of Reciprocal of Hyperbolic Sine of a x
:$\displaystyle \int \frac {\mathrm d x} {\sinh a x} = \frac 1 a \ln \left\vert {\tanh \frac {a x} 2} \right\vert + C$
Primitive of Reciprocal of Hyperbolic Tangent of a x
:$\displaystyle \int \frac {\mathrm d x} {\tanh a x} = \frac {\ln \left\vert {\sinh a x} \right\vert} a + C$
Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent
:$\coth x = \dfrac 1 {\tanh x}$ where $\tanh$ and $\coth$ denote hyperbolic tangent and hyperbolic cotangent respectively.
Primitive of Reciprocal of Hyperbolic Cotangent of a x
:$\displaystyle \int \frac {\mathrm d x} {\coth a x} = \frac {\ln \left\vert {\cosh a x} \right\vert} a + C$
Primitive of Reciprocal of Hyperbolic Cosecant of a x
:$\displaystyle \int \frac {\mathrm d x} {\operatorname{csch} a x} = \frac {\cosh a x} a + C$
Primitive of Square of Hyperbolic Sine of a x
:$\displaystyle \int \sinh^2 a x \rd x = \dfrac {\sinh a x \cosh a x} {2 a} - \frac x 2 + C$
Primitive of Square of Hyperbolic Cosine of a x
:$\displaystyle \int \cosh^2 a x \rd x = \frac {\sinh a x \cosh a x} {2 a} + \frac x 2 + C$
Primitive of Square of Hyperbolic Tangent of a x
:$\displaystyle \int \tanh^2 a x \rd x = x - \frac {\tanh a x} a + C$
Primitive of Square of Hyperbolic Cotangent of a x
:$\displaystyle \int \coth^2 a x \ \mathrm d x = x - \frac {\coth a x} a + C$
Primitive of Square of Hyperbolic Secant of a x
:$\ds \int \sech^2 a x \rd x = \frac {\tanh a x} a + C$
Primitive of Square of Hyperbolic Cosecant of a x
:$\displaystyle \int \operatorname{csch}^2 a x \ \mathrm d x = \frac {-\coth a x} a + C$
Primitive of Square of Hyperbolic Sine of a x/Corollary
:$\displaystyle \int \sinh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} - \frac x 2 + C$
Primitive of x by Square of Hyperbolic Sine of a x
:$\ds \int x \sinh^2 a x \rd x = \dfrac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} - \frac {x^2} 4 + C$
Primitive of Square of Hyperbolic Cosine of a x/Corollary
:$\displaystyle \int \cosh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} + \frac x 2 + C$
Primitive of x by Square of Hyperbolic Cosine of a x
:$\displaystyle \int x \cosh^2 a x \ \mathrm d x = \frac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} + \frac {x^2} 4 + C$
Primitive of x by Square of Hyperbolic Tangent of a x
:$\displaystyle \int x \tanh^2 a x \ \mathrm d x = \frac {x^2} 2 - \frac {x \tanh a x} a + \frac 1 {a^2} \ln \left\vert{\cosh a x}\right\vert + C$
Primitive of x by Square of Hyperbolic Cotangent of a x
:$\displaystyle \int x \coth^2 a x \ \mathrm d x = \frac {x^2} 2 - \frac {x \coth a x} a + \frac 1 {a^2} \ln \left\vert{\sinh a x}\right\vert + C$
Primitive of x by Square of Hyperbolic Secant of a x
:$\displaystyle \int x \operatorname{sech}^2 a x \ \mathrm d x = \frac {x \tanh a x} a - \frac 1 {a^2} \ln \left\vert{\cosh a x}\right\vert + C$
Primitive of Hyperbolic Sine of a x by Hyperbolic Sine of p x
:$\displaystyle \int \sinh a x \sinh p x \rd x = \frac {\map \sinh {a + p} x} {2 \paren {a + p} } - \frac {\map \sinh {a - p} x} {2 \paren {a - p} } + C$
Primitive of Hyperbolic Cosine of a x by Hyperbolic Cosine of p x
:$\displaystyle \int \cosh a x \cosh p x \rd x = \frac {\map \sinh {a + p} x} {2 \paren {a + p} } + \frac {\map \sinh {a - p} x} {2 \paren {a - p} } + C$
Primitive of Hyperbolic Sine of a x by Sine of p x
:$\displaystyle \int \sinh a x \sin p x \ \mathrm d x = \frac {a \cosh a x \sin p x - p \sinh a x \cos p x} {a^2 + p^2} + C$
Primitive of Hyperbolic Cosine of a x by Sine of p x
:$\displaystyle \int \cosh a x \sin p x \ \mathrm d x = \frac {a \sinh a x \sin p x - p \cosh a x \cos p x} {a^2 + p^2} + C$
Primitive of Hyperbolic Cosine of a x by Cosine of p x
:$\displaystyle \int \cosh a x \cos p x \ \mathrm d x = \frac {a \sinh a x \cos p x + p \cosh a x \sin p x} {a^2 + p^2} + C$
Primitive of Reciprocal of p plus q by Hyperbolic Sine of a x
:$\ds \int \frac {\d x} {p + q \sinh a x}  = \frac 1 {a \sqrt{p^2 + q^2} } \ln \size {\frac {q e^{a x} + p - \sqrt {p^2 + q^2} } {q e^{a x} + p + \sqrt {p^2 + q^2} } } + C$
Primitive of Reciprocal of q plus p by Hyperbolic Secant of a x
:$\displaystyle \int \frac {\mathrm d x} {q + p \operatorname{sech} a x} = \frac x q - \frac p q \int \frac {\mathrm d x} {p + q \cosh a x} + C$
Primitive of Reciprocal of q plus p by Hyperbolic Cosecant of a x
:$\displaystyle \int \frac {\d x} {q + p \csch a x} = \frac x q - \frac p q \int \frac {\d x} {p + q \sinh a x} + C$
Primitive of Power of x by Hyperbolic Cosine of a x
:$\ds \int x^m \cosh a x \rd x = \frac {x^m \sinh a x} a - \frac m a \int x^{m - 1} \sinh a x \rd x + C$
Primitive of Power of Hyperbolic Sine of a x
:$\displaystyle \int \sinh^n a x \rd x = \frac {\sinh^{n - 1} a x \cosh a x} {a n} - \frac {n - 1} n \int \sinh^{n - 2} a x \rd x + C$
Primitive of Power of Hyperbolic Cosine of a x
:$\displaystyle \int \cosh^n a x \rd x = \frac {\cosh^{n - 1} a x \sinh a x} {a n} + \frac {n - 1} n \int \cosh^{n - 2} a x \rd x + C$
Primitive of Power of Hyperbolic Tangent of a x by Square of Hyperbolic Secant of a x
:$\displaystyle \int \tanh^n a x \operatorname{sech}^2 a x \ \mathrm d x = \frac {\tanh^{n + 1} a x} {\left({n + 1}\right) a} + C$
Derivative of Hyperbolic Tangent of a x
:$\map {\dfrac \d {\d x} } {\tanh a x} = a \sech^2 a x$
Derivative of Hyperbolic Cotangent of a x
:$\map {\dfrac \d {\d x} } {\coth a x} = -a \csch^2 a x$
Primitive of Power of Hyperbolic Cotangent of a x by Square of Hyperbolic Cosecant of a x
:$\ds \int \coth^n a x \csch^2 a x \rd x = \frac {-\coth^{n + 1} a x} {\paren {n + 1} a} + C$
Primitive of Power of Hyperbolic Cotangent of a x
:$\ds \int \coth^n a x \rd x = \frac {-\coth^{n - 1} a x} {a \paren {n - 1} } + \int \coth^{n - 2} a x \rd x + C$
Derivative of Hyperbolic Secant of a x
:$\map {\dfrac \d {\d x} } {\sech a x} = -a \sech a x \tanh a x$
Derivative of Hyperbolic Cosecant of a x
:$\map {\dfrac \d {\d x} } {\csch a x} = -a \csch a x \coth a x$
Primitive of Power of Hyperbolic Secant of a x
:$\displaystyle \int \sech^n a x \rd x = \frac {\sech^{n - 2} a x \tanh a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sech^{n - 2} a x \rd x + C$
Primitive of Power of Hyperbolic Cosecant of a x
:$\displaystyle \int \csch^n a x \rd x = \frac {-\csch^{n - 2} a x \coth a x} {a \paren {n - 1} } - \frac {n - 2} {n - 1} \int\csch^{n - 2} a x \rd x + C$
Primitive of x over Power of Hyperbolic Sine of a x
:$\displaystyle \int \frac {x \rd x} {\sinh^n a x} = \frac {-x \cosh a x} {a \paren {n - 1} \sinh^{n - 1} a x} - \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \sinh^{n - 2} a x} - \frac {n - 2} {n - 1} \int \frac {x \rd x} {\sinh^{n - 2} a x} + C$
Primitive of x over Power of Hyperbolic Cosine of a x
:$\displaystyle \int \frac {x \rd x} {\cosh^n a x} = \frac {x \sinh a x} {a \paren {n - 1} \cosh^{n - 1} a x} + \frac 1 {a^2 \paren {n - 1} \paren {n - 2} \cosh^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {x \rd x} {\cosh^{n - 2} a x} + C$
Primitive of Reciprocal of Hyperbolic Cosine of a x plus 1
:$\ds \int \frac {\d x} {\cosh a x + 1} = \frac 1 a \tanh \frac {a x} 2 + C$
Primitive of Reciprocal of Square of Hyperbolic Cosine of a x minus 1
:$\displaystyle \int \frac {\d x} {\paren {\cosh a x - 1}^2} = \frac 1 {2 a} \coth \frac {a x} 2 - \frac 1 {6 a} \coth^3 \frac {a x} 2 + C$
Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x
:$\displaystyle \int \sinh a x \cosh a x \rd x = \frac {\sinh^2 a x} {2 a} + C$
Primitive of Power of Hyperbolic Sine of a x by Hyperbolic Cosine of a x
:$\displaystyle \int \sinh^n a x \cosh a x \rd x = \frac {\sinh^{n + 1} a x} {\paren {n + 1} a} + C$ for $n \ne -1$.
Primitive of Power of Hyperbolic Cosine of a x by Hyperbolic Sine of a x
:$\displaystyle \int \cosh^n a x \sinh a x \rd x = \frac {\cosh^{n + 1} a x} {\paren {n + 1} a} + C$ for $n \ne -1$.
Primitive of Square of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x
:$\displaystyle \int \sinh^2 a x \cosh^2 a x \rd x = \frac {\sinh 4 a x} {32 a} - \frac x 8 + C$
Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x
:$\displaystyle \int \frac {\mathrm d x} {\sinh a x \cosh a x} = \frac 1 a \ln \left\vert{\tanh a x}\right\vert + C$
Primitive of Square of Hyperbolic Secant of a x over Hyperbolic Tangent of a x
:$\displaystyle \int \frac {\operatorname{sech}^2 a x \ \mathrm d x} {\tanh a x} = \frac 1 a \ln \left\vert{\tanh a x}\right\vert + C$
Primitive of Reciprocal of Square of Hyperbolic Sine of a x by Hyperbolic Cosine of a x
:$\ds \int \frac {\d x} {\sinh^2 a x \cosh a x} = -\frac 1 a \map \arctan {\sinh a x} - \frac {\csch a x} a + C$
Primitive of Power of Hyperbolic Secant of a x by Hyperbolic Tangent of a x
:$\displaystyle \int \sech^n a x \tanh a x \rd x = \frac {-\sech^n a x} {n a} + C$
Primitive of Power of Hyperbolic Cosecant of a x by Hyperbolic Cotangent of a x
:$\displaystyle \int \csch^n a x \coth a x \rd x = \frac {-\csch^n a x} {n a} + C$
Primitive of Square of Hyperbolic Sine of a x over Hyperbolic Cosine of a x
:$\displaystyle \int \frac {\sinh^2 a x \ \mathrm d x} {\cosh a x} = \frac {\sinh a x} a - \frac 1 a \arctan \left({\sinh a x}\right) + C$
Primitive of Reciprocal of Hyperbolic Cosine of a x by 1 plus Hyperbolic Sine of a x
:$\ds \int \frac {\rd x} {\cosh a x \paren {1 + \sinh a x} } = \frac 1 {2 a} \ln \size {\frac {1 + \sinh a x} {\cosh a x} } + \frac 1 a \map \arctan {e^{a x} } + C$
Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x minus 1
:$\displaystyle \int \frac {\d x} {\sinh a x \paren {\cosh a x - 1} } = \frac {-1} {2 a} \ln \size {\tanh \frac {a x} 2} - \frac 1 {2 a \paren {\cosh a x - 1} } + C$
Primitive of Cube of Hyperbolic Tangent of a x
:$\displaystyle \int \tanh^3 a x \ \mathrm d x = \frac {\ln \left\vert{\cosh a x}\right\vert} a - \frac {\tanh^2 a x} {2 a} + C$
Primitive of Cube of Hyperbolic Tangent of a x/Proof 1
:$\displaystyle \int \tanh^3 a x \ \mathrm d x = \frac {\ln \left\vert{\cosh a x}\right\vert} a - \frac {\tanh^2 a x} {2 a} + C$
Primitive of Cube of Hyperbolic Tangent of a x/Proof 2
:$\displaystyle \int \tanh^3 a x \ \mathrm d x = \frac {\ln \left\vert{\cosh a x}\right\vert} a - \frac {\tanh^2 a x} {2 a} + C$
Primitive of Cube of Hyperbolic Cotangent of a x/Proof 1
:$\displaystyle \int \coth^3 a x \ \mathrm d x = \frac {\ln \left\vert{\sinh a x}\right\vert} a - \frac {\coth^2 a x} {2 a} + C$
Primitive of Cube of Hyperbolic Cotangent of a x/Proof 2
:$\displaystyle \int \coth^3 a x \ \mathrm d x = \frac {\ln \left\vert{\sinh a x}\right\vert} a - \frac {\coth^2 a x} {2 a} + C$
Primitive of x by Hyperbolic Tangent of a x
:$\displaystyle \int x \tanh a x \rd x = \frac 1 {a^2} \paren {\frac {\paren {a x}^3} 3 - \frac {\paren {a x}^5} {15} + \frac {2 \paren {a x}^7} {105} + \cdots + \frac { 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$
Primitive of x by Hyperbolic Cotangent of a x
:$\displaystyle \int x \coth a x \rd x = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} 9 - \frac {\paren {a x}^5} {225} + \cdots + \frac {2^{2 n} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$
Primitive of Square of Hyperbolic Cosecant of a x over Hyperbolic Cotangent of a x
:$\displaystyle \int \frac {\csch^2 a x \rd x} {\coth a x} = \frac {-1} a \ln \size {\coth a x} + C$
Primitive of Cube of Hyperbolic Secant of a x
:$\displaystyle \int \sech^3 a x \rd x = \frac {\sech a x \tanh a x} {2 a} + \frac 1 {2 a} \map \arctan {\sinh a x} + C$
Primitive of Cube of Hyperbolic Cosecant of a x
:$\ds \int \csch^3 a x \rd x = \frac {-\csch a x \coth a x} {2 a} - \frac 1 {2 a} \ln \size {\tanh a x} + C$
Primitive of Hyperbolic Secant Function/Arctangent of Hyperbolic Sine Form
:$\ds \int \sech x \rd x =  \map \arctan {\sinh x} + C$
Primitive of Hyperbolic Secant of a x/Arcsine Form
:$\displaystyle \int \operatorname{sech} a x \ \mathrm d x = \frac {\arcsin \left({\tanh a x}\right)} a + C$
Primitive of Hyperbolic Secant of a x/Arctangent of Exponential Form
:$\ds \int \sech a x \rd x = \frac {2 \map \arctan {e^{a x} } } a + C$
Primitive of Hyperbolic Secant of a x/Arctangent of Hyperbolic Sine Form
:$\displaystyle \int \operatorname{sech} a x \ \mathrm d x = \frac {\arctan \left({\sinh a x}\right)} a + C$
Primitive of x by Hyperbolic Cosecant of a x
:$\displaystyle \int x \csch a x \rd x = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + C$
Derivative of Inverse Hyperbolic Sine of x over a
:$\dfrac {\map \d {\map {\sinh^{-1} } {\frac x a} } } {\d x} = \dfrac 1 {\sqrt {x^2 + a^2}}$
Primitive of Inverse Hyperbolic Sine of x over a
:$\displaystyle \int \sinh^{-1} \frac x a \rd x = x \sinh^{-1} \frac x a - \sqrt {x^2 + a^2} + C$
Derivative of Inverse Hyperbolic Cosine of x over a
:$\dfrac {\map \d {\map {\cosh^{-1} } {\frac x a} } } {\d x} = \dfrac 1 {\sqrt {x^2 - a^2} }$ where $x > a$.
Derivative of Inverse Hyperbolic Tangent of x over a
:$\map {\dfrac \d {\d x} } {\map {\tanh^{-1} } {\dfrac x a} } = \dfrac a {a^2 - x^2}$ where $-a < x < a$.
Derivative of Inverse Hyperbolic Cotangent of x over a
:$\map {\dfrac \d {\d x} } {\map {\coth^{-1} } {\dfrac x a} } = \dfrac {-a} {x^2 - a^2}$ where $x^2 > a^2$.
Derivative of Inverse Hyperbolic Secant of x over a
:$\dfrac {\mathrm d \left({\operatorname{sech}^{-1} \left({\frac x a}\right)}\right)} {\mathrm d x} = \dfrac {-a} {x \sqrt{a^2 - x^2} }$ where $0 < x < a$.
Derivative of Inverse Hyperbolic Cosecant of x over a
:$\dfrac {\map \d {\csch^{-1} } {\frac x a} } {\d x} = \dfrac {-a} {\size x \sqrt {a^2 + x^2} }$ where $x \ne 0$.
Primitive of Inverse Hyperbolic Tangent of x over a
:$\displaystyle \int \tanh^{-1} \frac x a \rd x = x \tanh^{-1} \dfrac x a + \frac {a \ln \left({a^2 - x^2}\right)} 2 + C$
Primitive of Inverse Hyperbolic Cotangent of x over a
:$\displaystyle \int \coth^{-1} \frac x a \ \mathrm d x = x \coth^{-1} \dfrac x a + \frac {a \ln \left({x^2 - a^2}\right)} 2 + C$
Primitive of Inverse Hyperbolic Secant of x over a
:$\displaystyle \int \sech^{-1} \frac x a \rd x = \begin{cases} x \sech^{-1} \dfrac x a + a \arcsin \dfrac x a + C & : \sech^{-1} \dfrac x a > 0 \\ x \sech^{-1} \dfrac x a - a \arcsin \dfrac x a + C & : \sech^{-1} \dfrac x a < 0 \end{cases}$
Inverse Hyperbolic Cosine of x over a in Logarithm Form
:$\cosh^{-1} \dfrac x a = \map \ln {x + \sqrt {x^2 - a^2} } - \ln a$
Inverse Hyperbolic Tangent of x over a in Logarithm Form
:$\tanh^{-1} \dfrac x a = \dfrac 1 2 \map \ln {\dfrac {a + x} {a - x} }$
Inverse Hyperbolic Cotangent of x over a in Logarithm Form
:$\coth^{-1} \dfrac x a = \dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} }$
Inverse Hyperbolic Secant of x over a in Logarithm Form
:$\operatorname{sech}^{-1} \dfrac x a = \ln \left({\dfrac {a + \sqrt{a^2 - x^2} } x}\right)$
Inverse Hyperbolic Cosecant of x over a in Logarithm Form
:$\csch^{-1} \dfrac x a = \map \ln {\dfrac {a + \sqrt {a^2 + x^2} } x}$
Primitive of Root of x squared plus a squared/Inverse Hyperbolic Sine Form
:$\displaystyle \int \sqrt {x^2 + a^2} \rd x = \frac {x \sqrt {x^2 + a^2} } 2 + \frac {a^2} 2 \sinh^{-1} \frac x a + C$
Primitive of x squared over Root of x squared plus a squared/Inverse Hyperbolic Sine Form
:$\displaystyle \int \frac {x^2 \rd x} {\sqrt {x^2 + a^2} } = \frac {x \sqrt {x^2 + a^2} } 2 - \frac {a^2} 2 \sinh^{-1} \frac x a + C$
Primitive of Root of x squared minus a squared/Inverse Hyperbolic Cosine Form
:$\displaystyle \int \sqrt {x^2 - a^2} \rd x = \frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \cosh^{-1} \frac x a + C$
Primitive of Root of x squared minus a squared/Logarithm Form
:$\displaystyle \int \sqrt {x^2 - a^2} \rd x = \frac {x \sqrt {x^2 - a^2} } 2 - \frac {a^2} 2 \ln \size {x + \sqrt {x^2 - a^2} } + C$
Primitive of x squared over Root of x squared minus a squared/Inverse Hyperbolic Cosine Form
:$\displaystyle \int \frac {x^2 \rd x} {\sqrt {x^2 - a^2} } = \frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \cosh^{-1} \frac x a + C$ for $x > a$.
Primitive of x squared over x squared minus a squared/Inverse Hyperbolic Cotangent Form
:$\displaystyle \int \frac {x^2 \ \mathrm d x} {x^2 - a^2} = x - a \coth^{-1} \frac x a + C$ for $x^2 > a^2$.
Primitive of x squared over a squared minus x squared/Logarithm Form
:$\displaystyle \int \frac {x^2 \rd x} {a^2 - x^2} = -x + \frac a 2 \map \ln {\frac {a + x} {a - x} } + C$ for $x^2 < a^2$.
Primitive of x squared over a squared minus x squared/Inverse Hyperbolic Tangent Form
:$\displaystyle \int \frac {x^2 \rd x} {a^2 - x^2} = -x + a \tanh^{-1} \frac x a + C$ for $x^2 < a^2$.
Primitive of x by Inverse Hyperbolic Tangent of x over a
:$\displaystyle \int x \tanh^{-1} \frac x a \rd x = \frac {a x} 2 + \frac {x^2 - a^2} 2 \tanh^{-1} \frac x a + C$
Primitive of x squared by Inverse Hyperbolic Sine of x over a
:$\displaystyle \int x^2 \sinh^{-1} \frac x a \rd x = \frac {x^3} 3 \sinh^{-1} \frac x a + \frac {\paren {2 a^2 - x^2} \sqrt {x^2 + a^2} } 9 + C$
Primitive of x squared by Inverse Hyperbolic Tangent of x over a
:$\displaystyle \int x^2 \tanh^{-1} \frac x a \ \mathrm d x = \frac {a x^2} 6 + \frac {x^3} 3 \tanh^{-1} \frac x a + \frac {a^3} 6 \ln \left({a^2 - x^2}\right) + C$
Primitive of Inverse Hyperbolic Sine of x over a over x squared
:$\displaystyle \int \frac {\sinh^{-1} \dfrac x a \ \mathrm d x} {x^2} = \frac {-\sinh^{-1} \dfrac x a} x - \frac 1 a \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right)$
Primitive of Inverse Hyperbolic Cosine of x over a over x squared
:$\displaystyle \int \frac {\cosh^{-1} \dfrac x a \ \mathrm d x} {x^2} = \begin{cases} \displaystyle \frac {-\cosh^{-1} \dfrac x a} x - \frac 1 a \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right) & : \cosh^{-1} \dfrac x a > 0 \\ \displaystyle \frac {-\cosh^{-1} \dfrac x a} x + \frac 1 a \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right) & : \cosh^{-1} \dfrac x a < 0 \\ \end{cases}$
Primitive of Inverse Hyperbolic Tangent of x over a over x squared
:$\ds \int \frac {\tanh^{-1} \dfrac x a \rd x} {x^2} = \frac {-\tanh^{-1} \dfrac x a} x + \frac 1 {2 a} \map \ln {\frac {x^2} {a^2 - x^2} } + C$
Primitive of Inverse Hyperbolic Cotangent of x over a over x squared
:$\displaystyle \int \frac {\coth^{-1} \dfrac x a \rd x} {x^2} = \frac {-\coth^{-1} \dfrac x a} x + \frac 1 {2 a} \map \ln {\frac {x^2} {x^2 - a^2} } + C$