hcju/proofwikiqa huggingface dataset:
From Primitive of $\dfrac {x^m} {\ln x}$: :$\displaystyle \int \frac {x^m \rd x} {\ln x} = \map \ln {\ln x} + \paren {m + 1} \ln x + \sum_{k \mathop \ge 2}^n \frac {\paren {m + 1}^k \paren {\ln x}^k} {k \times k!} + C$ The result follows by setting $m = 1$. {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \ln^n x       | c =  }} {{eqn | ll= \implies       | l = \frac {\d u} {\d x}       | r = n \ln^{n - 1} x \frac 1 x       | c = Derivative of $\ln x$, Derivative of Power, Chain Rule for Derivatives }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = 1       | c =  }} {{eqn | ll= \implies       | l = v       | r = x       | c = Primitive of Constant }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \ln^n x \rd x       | r = x \ln^n x - \int x \left({n \ln^{n - 1} x \frac 1 x}\right) \rd x + C       | c = Integration by Parts }} {{eqn | r = x \ln^n x - n \int \ln^{n - 1} x \rd x + C       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$ let: {{begin-eqn}} {{eqn | l = u       | r = \ln^n x       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u} {\mathrm d x}       | r = n \ln^{n - 1} x \frac 1 x       | c = Derivative of $\ln x$, Derivative of Power, Chain Rule for Derivatives }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v} {\mathrm d x}       | r = x^m       | c =  }} {{eqn | ll= \implies       | l = v       | r = \frac {x^{m + 1} }{m + 1}       | c = Primitive of Power }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x^m \ln^n x \ \mathrm d x       | r = \frac {x^{m + 1} }{m + 1} \ln^n x - \int \frac {x^{m + 1} }{m + 1} \left({n \ln^{n - 1} x \frac 1 x}\right) \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^{m + 1} \ln^n x} {m + 1} - \frac n {m + 1} \int x^m \ln^{n - 1} x \ \mathrm d x + C       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \map \ln {x^2 + a^2}       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = \frac {2 x} {x^2 + a^2}       | c = Derivative of $\ln x$, Derivative of Power, Chain Rule for Derivatives }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = 1       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = x       | c = Primitive of Power }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \map \ln {x^2 + a^2} \rd x       | r = x \map \ln {x^2 + a^2} - \int \frac {2 x^2 \rd x} {x^2 + a^2} + C       | c = Integration by Parts }} {{eqn | r = x \map \ln {x^2 + a^2} - 2 \int \frac {x^2 \rd x} {x^2 + a^2} + C       | c = Primitive of Constant Multiple of Function }} {{eqn | r = x \map \ln {x^2 + a^2} - 2 \paren {x - a \arctan {\frac x a} } + C       | c = Primitive of $\dfrac {x^2} {x^2 + a^2}$ }} {{eqn | r = x \map \ln {x^2 + a^2} - 2 x + 2 a \arctan \frac x a + C       | c = simplifying }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \ln \left({x^2 - a^2}\right)       | c =  }} {{eqn | ll= \implies       | l = \frac {\d u} {\d x}       | r = \frac {2 x} {x^2 - a^2}       | c = Derivative of $\ln x$, Derivative of Power, Chain Rule for Derivatives }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = 1       | c =  }} {{eqn | ll= \implies       | l = v       | r = x       | c = Primitive of Power }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \ln \left({x^2 - a^2}\right) \rd x       | r = x \ln \left({x^2 - a^2}\right) - \int \frac {2 x^2 \rd x} {x^2 - a^2} + C       | c = Integration by Parts }} {{eqn | r = x \ln \left({x^2 - a^2}\right) - 2 \int \frac {x^2 \rd x} {x^2 - a^2} + C       | c = Primitive of Constant Multiple of Function }} {{eqn | r = x \ln \left({x^2 - a^2}\right) - 2 \left({x + \frac a 2 \ln \left({\frac {x - a} {x + a} }\right)}\right) + C       | c = Primitive of $\dfrac {x^2} {x^2 - a^2}$ }} {{eqn | r = x \ln \left({x^2 - a^2}\right) - 2 x - a \ln \left({\frac {x - a} {x + a} }\right) + C       | c = simplifying }} {{eqn | r = x \ln \left({x^2 - a^2}\right) - 2 x + a \ln \left({\frac {x + a} {x - a} }\right) + C       | c = Logarithm of Reciprocal }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \map \ln {x^2 + a^2}       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = \frac {2 x} {x^2 + a^2}       | c = Derivative of $\ln x$, Derivative of Power, Chain Rule for Derivatives }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = x^m       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {x^{m + 1} } {m + 1}       | c = Primitive of Power }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x^m \map \ln {x^2 + a^2} \rd x       | r = \frac {x^{m + 1} } {m + 1} \map \ln {x^2 + a^2} - \int \frac {x^{m + 1} } {m + 1} \frac {2 x \rd x} {x^2 + a^2} + C       | c = Integration by Parts }} {{eqn | r = \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} \d x       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \map \ln {x^2 - a^2}       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = \frac {2 x} {x^2 - a^2}       | c = Derivative of $\ln x$, Derivative of Power, Chain Rule for Derivatives }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = x^m       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {x^{m + 1} } {m + 1}       | c = Primitive of Power }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x^m \, \map \ln {x^2 - a^2} \rd x       | r = \frac {x^{m + 1} } {m + 1} \map \ln {x^2 - a^2} - \int \frac {x^{m + 1} } {m + 1} \frac {2 x \rd x} {x^2 - a^2} + C       | c = Integration by Parts }} {{eqn | r = \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       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \sinh x \ \mathrm d x       | r = \cosh x + C       | c = Primitive of $\sinh x$ }} {{eqn | ll= \implies       | l = \int \sinh a x \ \mathrm d x       | r = \frac 1 a \left({\cosh a x}\right) + C       | c = Primitive of Function of Constant Multiple }} {{eqn | r = \frac {\cosh a x} a + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \sinh a x \cosh a x \rd x       | r = \int \frac {\sinh 2 a x} 2 \rd x       | c = Double Angle Formula for Hyperbolic Sine }} {{eqn | r = \frac 1 2 \int \sinh 2 a x \rd x       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 2 \paren {\frac {\cosh 2 a x} {2 a} } + C       | c = Primitive of $\sinh a x$ }} {{eqn | r = \frac 1 2 \paren {\frac {1 + 2 \sinh^2 a x} {2 a} } + C       | c = Corollary to Double Angle Formula for Hyperbolic Cosine }} {{eqn | r = \frac {\sinh^2 a x} {2 a} + \frac 1 {4 a} + C       | c = simplifying }} {{eqn | r = \frac {\sinh^2 a x} {2 a} + C       | c = subsuming $\dfrac 1 {4 a}$ into arbitrary constant }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | n = 1       | l = \int \sinh^n a x \cosh a x \rd x       | r = \frac {\sinh^{n + 1} a x} {\paren {n + 1} a} + C       | c = Primitive of $\sinh^n a x \cosh a x$ }} {{eqn | ll= \leadsto       | l = \int \sinh a x \cosh a x \rd x       | r = \frac {\sinh^2 a x} {2 a} + C       | c = setting $n = 1$ in $(1)$ }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \sinh a x       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = a \cosh a x       | c = Derivative of $\sinh a x$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = \cosh a x       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {\sinh a x} a       | c = Primitive of $\cosh a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \sinh a x \cosh a x \rd x       | r = \paren {\sinh a x} \paren {\frac {\sinh a x} a} - \int \paren {\frac {\sinh a x} a} \paren {a \cosh a x} \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {\sinh^2 a x} a - \int \sinh a x \cosh a x \rd x + C       | c = simplifying }} {{eqn | ll= \leadsto       | l = 2 \int \sinh a x \cosh a x \rd x       | r = \frac {\sinh^2 a x} a + C       | c = gathering terms }} {{eqn | ll= \leadsto       | l = \int \sinh a x \cosh a x \rd x       | r = \frac {\sinh^2 a x} {2 a} + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = u       | r = \sinh a x       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = a \cosh a x       | c = Derivative of $\sinh a x$ }} {{eqn | ll= \leadsto       | l = \int \sinh a x \cosh a x \rd x       | r = \int \frac u a \rd u       | c = Integration by Substitution }} {{eqn | r = \frac 1 a \int u \rd u       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 a \frac {u^2} 2 + C       | c = Primitive of Power }} {{eqn | r = \frac {\sinh^2 a x} {2 a} + C       | c = substituting for $u$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \sinh a x \cosh a x \rd x       | r = \int \cosh a x \sinh a x \rd x       | c =  }} {{eqn | r = \frac {\cosh^2 a x} {2 a} + C       | c = Primitive of $\cosh^n a x \sinh a x$ using $n = 1$ }} {{eqn | r = \frac {1 + \sinh^2 a x} {2 a} + C       | c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn | r = \frac 1 {2 a} + \frac {\sinh^2 a x} {2 a} + C       | c = simplifying }} {{eqn | r = \frac {\sinh^2 a x} {2 a} + C       | c = subsuming $\dfrac 1 {2 a}$ into arbitrary constant }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \cosh x \ \mathrm d x       | r = \sinh x + C       | c = Primitive of $\cosh x$ }} {{eqn | ll= \implies       | l = \int \cosh a x \ \mathrm d x       | r = \frac 1 a \left({\sinh a x}\right) + C       | c = Primitive of Function of Constant Multiple }} {{eqn | r = \frac {\sinh a x} a + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \tanh x \rd x       | r = \ln \size {\cosh x} + C       | c = Primitive of $\tanh x$ }} {{eqn | ll= \leadsto       | l = \int \tanh a x \rd x       | r = \frac 1 a \paren {\ln \size {\cosh a x} } + C       | c = Primitive of Function of Constant Multiple }} {{eqn | r = \frac {\ln \size {\cosh a x} } a + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \coth x \ \mathrm d x       | r = \ln \left \vert {\sinh x} \right \vert + C       | c = Primitive of $\coth x$ }} {{eqn | ll= \implies       | l = \int \coth a x \ \mathrm d x       | r = \frac 1 a \left({\ln \left \vert {\sinh a x} \right \vert}\right) + C       | c = Primitive of Function of Constant Multiple }} {{eqn | r = \frac {\ln \left\vert{\sinh a x}\right\vert} a + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \csch x \rd x       | r = \ln \size {\tanh \frac x 2} + C       | c = Primitive of $\csch x$ }} {{eqn | ll= \leadsto       | l = \int \csch a x \rd x       | r = \frac 1 a \ln \size {\tanh \frac {a x} 2} + C       | c = Primitive of Function of Constant Multiple }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$ let: {{begin-eqn}} {{eqn | l = u       | r = x       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u}{\mathrm d x}       | r = 1       | c = Derivative of Identity Function }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v}{\mathrm d x}       | r = \sinh a x       | c =  }} {{eqn | ll= \implies       | l = v       | r = \frac {\cosh a x} a       | c = Primitive of $\sinh a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x \sinh a x \ \mathrm d x       | r = x \left({\frac {\cosh a x} a}\right) - \int \left({\frac {\cosh a x} a}\right) \times 1 \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = \frac {x \cosh a x} a - \frac 1 a \int \cosh a x \ \mathrm d x + C       | c = Linear Combination of Integrals }} {{eqn | r = \frac {x \cosh a x} a - \frac 1 a \left({\frac {\sinh a x} a}\right) + C       | c = Primitive of $\cosh a x$ }} {{eqn | r = \frac {x \cosh a x} a - \frac {\sinh a x} {a^2} + C       | c = simplification }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$ 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 = \sinh a x       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {\sinh a x} a       | c = Primitive of $\cosh a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x \cosh a x \rd x       | r = x \paren {\frac {\sinh a x} a} - \int \paren {\frac {\sinh a x} a} \times 1 \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {x \sinh a x} a - \frac 1 a \int \sinh a x \rd x + C       | c = Linear Combination of Integrals }} {{eqn | r = \frac {x \sinh a x} a - \frac 1 a \paren {\frac {\cosh a x} a} + C       | c = Primitive of $\sinh a x$ }} {{eqn | r = \frac {x \sinh a x} a - \frac {\cosh a x} {a^2} + C       | c = simplification }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$ let: {{begin-eqn}} {{eqn | l = u       | r = x^2       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u}{\mathrm d x}       | r = 2 x       | c = Derivative of Power }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v}{\mathrm d x}       | r = \sinh a x       | c =  }} {{eqn | ll= \implies       | l = v       | r = \frac {\cosh a x} a       | c = Primitive of $\sinh a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x^2 \sinh a x \ \mathrm d x       | r = x^2 \left({\frac {\cosh a x} a}\right) - \int 2 x \left({\frac {\cosh a x} a}\right) \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^2 \cosh a x} a - \frac 2 a \int x \cosh a x \ \mathrm d x + C       | c = Linear Combination of Integrals }} {{eqn | r = \frac {x^2 \cosh a x} a - \frac 2 a \left({\frac {x \sinh a x} a - \frac {\cosh a x} {a^2} }\right) + C       | c = Primitive of $x \cosh a x$ }} {{eqn | r = \left({\frac {x^2} a + \frac 2 {a^3} }\right) \cosh a x - \frac {2 x \sinh a x} {a^2} + C       | c = simplification }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$ let: {{begin-eqn}} {{eqn | l = u       | r = x^2       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u}{\mathrm d x}       | r = 2 x       | c = Derivative of Power }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v}{\mathrm d x}       | r = \cosh a x       | c =  }} {{eqn | ll= \implies       | l = v       | r = \frac {\sinh a x} a       | c = Primitive of $\cosh a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x^2 \sinh a x \ \mathrm d x       | r = x^2 \left({\frac {\sinh a x} a}\right) - \int 2 x \left({\frac {\sinh a x} a}\right) \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^2 \sinh a x} a - \frac 2 a \int x \sinh a x \ \mathrm d x + C       | c = Linear Combination of Integrals }} {{eqn | r = \frac {x^2 \sinh a x} a - \frac 2 a \left({\frac {x \cosh a x} a - \frac {\sinh a x} {a^2} }\right) + C       | c = Primitive of $x \cosh a x$ }} {{eqn | r = \left({\frac {x^2} a + \frac 2 {a^3} }\right) \sinh a x - \frac {2 x \cosh a x} {a^2} + C       | c = simplification }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\cosh a x \ \mathrm d x} x       | r = \int \frac 1 x \left({\sum_{k \mathop = 0}^\infty \frac {\left({a x}\right)^{2 k} }{\left({2 k}\right)!} }\right) \ \mathrm d x       | c = Taylor Series Expansion for Hyperbolic Cosine Function }} {{eqn | r = \sum_{k \mathop = 0}^\infty \frac {a^{2 k} } {\left({2 k}\right)!} \int \frac 1 x \left({x^{2 k} }\right) \ \mathrm d x       | c = Linear Combination of Integrals }} {{eqn | n = 1       | r = \int \frac {\mathrm d x} x + \sum_{k \mathop = 1}^\infty \frac {a^{2 k} } {\left({2 k}\right)!} \int \left({x^{2 k - 1} }\right) \ \mathrm d x       | c = extracting case for $k = 0$ }} {{eqn | r = \int \frac {\mathrm d x} x + \sum_{k \mathop = 1}^\infty \frac {a^{2 k} } {\left({2 k}\right)!} \frac {x^{2 k} } {2 k} + C       | c = Primitive of Power }} {{eqn | r = \int \frac {\mathrm d x} x + \sum_{k \mathop \ge 1} \frac {\left({a x}\right)^{2 k} } {\left({2 k}\right) \left({2 k}\right)!} + C       | c = simplifying }} {{eqn | 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       | c = Primitive of Reciprocal }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {D_x} {\sinh x}       | r = \cosh x       | c = Derivative of $\sinh x$ }} {{eqn | ll= \leadsto       | l = \map {D_x} {\sinh a x}       | r = a \cosh a x       | c = Derivative of Function of Constant Multiple }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\dfrac \d {\d x} } {\cosh x}       | r = \sinh x       | c = Derivative of $\cosh x$ }} {{eqn | ll= \leadsto       | l = \map {\dfrac \d {\d x} } {\cosh a x}       | r = a \sinh a x       | c = Derivative of Function of Constant Multiple }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$ let: {{begin-eqn}} {{eqn | l = u       | r = \sinh a x       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u} {\mathrm d x}       | r = a \cosh a x       | c = Derivative of $\sinh a x$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v} {\mathrm d x}       | r = \frac 1 {x^2}       | c =  }} {{eqn | ll= \implies       | l = v       | r = - \frac 1 x       | c = Primitive of Power }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \frac {\sinh a x \ \mathrm d x} {x^2}       | r = \int \sinh a x \frac 1 {x^2} \ \mathrm d x       | c =  }} {{eqn | r = \sinh a x \left({- \frac 1 x}\right) - \int \left({- \frac 1 x}\right) a \cosh a x \ \mathrm d x       | c = Integration by Parts }} {{eqn | r = -\frac {\sinh a x} x + a \int \frac {\cosh a x \ \mathrm d x} x       | c = simplifying }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$ let: {{begin-eqn}} {{eqn | l = u       | r = \cosh a x       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u} {\mathrm d x}       | r = a \sinh a x       | c = Derivative of $\cosh a x$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v} {\mathrm d x}       | r = \frac 1 {x^2}       | c =  }} {{eqn | ll= \implies       | l = v       | r = - \frac 1 x       | c = Primitive of Power }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \frac {\cosh a x \ \mathrm d x} {x^2}       | r = \int \cosh a x \frac 1 {x^2} \ \mathrm d x       | c =  }} {{eqn | r = \cosh a x \left({- \frac 1 x}\right) - \int \left({- \frac 1 x}\right) a \sinh a x \ \mathrm d x       | c = Integration by Parts }} {{eqn | r = -\frac {\cosh a x} x + a \int \frac {\sinh a x \ \mathrm d x} x       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\mathrm d x} {\sinh a x}       | r = \int \operatorname{csch} a x \ \mathrm d x       | c = Definition of Hyperbolic Cosecant }} {{eqn | r = \frac 1 a \ln \left\vert {\tanh \frac {a x} 2} \right\vert + C       | c = Primitive of $\operatorname{csch} a x$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\d x} {\cosh a x}       | r = \int \sech a x \rd x       | c = {{Defof|Hyperbolic Secant|index = 2}} }} {{eqn | r = \frac {2 \map \arctan {e^{a x} } } a + C       | c = Primitive of $\sech a x$: Arctangent of Exponential Form }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\mathrm d x} {\tanh a x}       | r = \int \coth a x \ \mathrm d x       | c = Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent }} {{eqn | r = \frac {\ln \left\vert {\sinh a x} \right\vert} a + C       | c = Primitive of $\coth a x$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\mathrm d x} {\coth a x}       | r = \int \tanh a x \ \mathrm d x       | c = Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent }} {{eqn | r = \frac {\ln \left\vert {\cosh a x} \right\vert} a + C       | c = Primitive of $\tanh a x$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\mathrm d x} {\operatorname{sech} a x}       | r = \int \cosh a x \ \mathrm d x       | c = Definition of Hyperbolic Secant }} {{eqn | r = \frac {\sinh a x} a + C       | c = Primitive of $\cosh a x$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\mathrm d x} {\operatorname{csch} a x}       | r = \int \sinh a x \ \mathrm d x       | c = Definition of Hyperbolic Cosecant }} {{eqn | r = \frac {\cosh a x} a + C       | c = Primitive of $\sinh a x$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \sinh^2 x \rd x       | r = \frac {\sinh x \cosh x - x} 2 + C       | c = Primitive of $\sinh^2 x$ }} {{eqn | ll= \leadsto       | l = \int \sinh^2 a x \rd x       | r = \frac 1 a \paren {\frac {\sinh a x \cosh a x - a x} 2} + C       | c = Primitive of Function of Constant Multiple }} {{eqn | r = \frac {\sinh a x \cosh a x} {2 a} - \frac x 2 + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \cosh^2 x \rd x       | r = \frac {\sinh x \cosh x + x} 2 + C       | c = Primitive of $\cosh^2 x$ }} {{eqn | ll= \implies       | l = \int \cosh^2 a x \rd x       | r = \frac 1 a \paren {\frac {\sinh a x \cosh a x + a x} 2} + C       | c = Primitive of Function of Constant Multiple }} {{eqn | r = \frac {\sinh a x \cosh a x} {2 a} + \frac x 2 + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \tanh^2 x \rd x       | r = x - \tanh x + C       | c = Primitive of $\tanh^2 x$ }} {{eqn | ll= \leadsto       | l = \int \tanh^2 a x \rd x       | r = \frac 1 a \paren {a x - \tanh a x} + C       | c = Primitive of Function of Constant Multiple }} {{eqn | r = x - \frac {\tanh a x} a + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \coth^2 x \ \mathrm d x       | r = x - \coth x + C       | c = Primitive of $\coth^2 x$ }} {{eqn | ll= \implies       | l = \int \coth^2 a x \ \mathrm d x       | r = \frac 1 a \left({a x - \coth a x}\right) + C       | c = Primitive of Function of Constant Multiple }} {{eqn | r = x - \frac {\coth a x} a + C       | c = simplifying }} {{end-eqn}} {{qed}}