hcju/proofwikiqa huggingface dataset:
{{begin-eqn}} {{eqn | l = \int \sech^2 x \rd x       | r = \tanh x + C       | c = Primitive of $\sech^2 x$ }} {{eqn | ll= \leadsto       | l = \int \sech^2 a x \rd x       | r = \frac 1 a \paren {\tanh a x} + C       | c = Primitive of Function of Constant Multiple }} {{eqn | r = \frac {\tanh a x} a + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \operatorname{csch}^2 x \ \mathrm d x       | r = -\coth x + C       | c = Primitive of $\operatorname{csch}^2 x$ }} {{eqn | ll= \implies       | l = \int \operatorname{sech}^2 a x \ \mathrm d x       | r = \frac 1 a \left({-\coth a x}\right) + C       | c = Primitive of Function of Constant Multiple }} {{eqn | r = \frac {-\coth a x} a + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \sinh^2 a x \rd x       | r = \frac {\sinh a x \cosh a x} {2 a} - \frac x 2 + C       | c = Primitive of $\sinh^2 a x$ }} {{eqn | r = \frac {\frac {\sinh 2 a x} 2} {2 a} - \frac x 2 + C       | c = Double Angle Formula for Hyperbolic Sine }} {{eqn | r = \dfrac {\sinh 2 a x} {4 a} - \frac x 2 + C       | c = simplifying }} {{end-eqn}} {{qed}}
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 = \sinh^2 a x       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {\sinh 2 a x} {4 a} - \frac x 2       | c = Primitive of $\sinh^2 a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x \sinh^2 a x \rd x       | r = x \paren {\frac {\sinh 2 a x} {4 a} - \frac x 2} - \int \paren {\frac {\sinh 2 a x} {4 a} - \frac x 2} \times 1 \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {x \sinh 2 a x} {4 a} - \frac {x^2} 2 - \frac 1 {4 a} \int \sinh 2 a x \rd x + \frac 1 2 \int x \rd x + C       | c = Linear Combination of Integrals }} {{eqn | r = \frac {x \sinh 2 a x} {4 a} - \frac {x^2} 2 - \frac 1 {4 a} \int \sinh 2 a x \rd x + \frac {x^2} 4 + C       | c = Primitive of Power }} {{eqn | r = \frac {x \sinh 2 a x} {4 a} - \frac {x^2} 2 - \frac 1 {4 a} \frac {\cosh 2 a x} {2 a} + \frac {x^2} 4 + C       | c = Primitive of $\sinh a x$ }} {{eqn | r = \dfrac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} - \frac {x^2} 4 + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \sinh^2 a x \rd x       | r = \frac {\sinh a x \cosh a x} {2 a} + \frac x 2 + C       | c = Primitive of $\cosh^2 a x$ }} {{eqn | r = \frac {\frac {\sinh 2 a x} 2} {2 a} + \frac x 2 + C       | c = Double Angle Formula for Hyperbolic Sine }} {{eqn | r = \dfrac {\sinh 2 a x} {4 a} + \frac x 2 + C       | 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 = 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 = \cosh^2 a x       | c =  }} {{eqn | ll= \implies       | l = v       | r = \frac {\sinh 2 a x} {4 a} + \frac x 2       | c = Primitive of $\cosh^2 a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x \cosh^2 a x \ \mathrm d x       | r = x \left({\frac {\sinh 2 a x} {4 a} + \frac x 2}\right) - \int \left({\frac {\sinh 2 a x} {4 a} + \frac x 2}\right) \times 1 \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = \frac {x \sinh 2 a x} {4 a} + \frac {x^2} 2 - \frac 1 {4 a} \int \sinh 2 a x \ \mathrm d x - \frac 1 2 \int x \ \mathrm d x + C       | c = Linear Combination of Integrals }} {{eqn | r = \frac {x \sinh 2 a x} {4 a} + \frac {x^2} 2 - \frac 1 {4 a} \int \sinh 2 a x \ \mathrm d x - \frac {x^2} 4 + C       | c = Primitive of Power }} {{eqn | r = \frac {x \sinh 2 a x} {4 a} + \frac {x^2} 2 - \frac 1 {4 a} \frac {\cosh 2 a x} {2 a} - \frac {x^2} 4 + C       | c = Primitive of $\sinh a x$ }} {{eqn | r = \frac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} + \frac {x^2} 4 + C       | 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 = 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 = \tanh^2 a x       | c =  }} {{eqn | ll= \implies       | l = v       | r = x - \frac {\tanh a x} a       | c = Primitive of $\tanh^2 a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x \tanh^2 a x \ \mathrm d x       | r = x \left({x - \frac {\tanh a x} a}\right) - \int \left({x - \frac {\tanh a x} a}\right) \times 1 \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = x^2 - \frac {x \tanh a x} a + \int x \ \mathrm d x + \frac 1 a \int \tanh a x \ \mathrm d x + C       | c = Linear Combination of Integrals }} {{eqn | r = x^2 - \frac {x \tanh a x} a + \frac {x^2} 2 + \frac 1 a \int \tanh a x \ \mathrm d x + C       | c = Primitive of Power }} {{eqn | r = x^2 - \frac {x \tanh a x} a + \frac {x^2} 2 + \frac 1 a \frac {\ln \left\vert{\cosh a x}\right\vert} a + C       | c = Primitive of $\tanh a x$ }} {{eqn | r = \frac {x^2} 2 - \frac {x \tanh a x} a + \frac 1 {a^2} \ln \left\vert{\cosh a x}\right\vert + C       | 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 = 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 = \coth^2 a x       | c =  }} {{eqn | ll= \implies       | l = v       | r = x - \frac {\coth a x} a       | c = Primitive of $\coth^2 a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x \coth^2 a x \ \mathrm d x       | r = x \left({x - \frac {\coth a x} a}\right) - \int \left({x - \frac {\coth a x} a}\right) \times 1 \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = x^2 - \frac {x \coth a x} a + \int x \ \mathrm d x + \frac 1 a \int \coth a x \ \mathrm d x + C       | c = Linear Combination of Integrals }} {{eqn | r = x^2 - \frac {x \coth a x} a + \frac {x^2} 2 + \frac 1 a \int \coth a x \ \mathrm d x + C       | c = Primitive of Power }} {{eqn | r = x^2 - \frac {x \coth a x} a + \frac {x^2} 2 + \frac 1 a \frac {\ln \left\vert{\sinh a x}\right\vert} a + C       | c = Primitive of $\coth a x$ }} {{eqn | r = \frac {x^2} 2 - \frac {x \coth a x} a + \frac 1 {a^2} \ln \left\vert{\sinh a x}\right\vert + C       | 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 = 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 = \operatorname{sech}^2 a x       | c =  }} {{eqn | ll= \implies       | l = v       | r = \frac {\tanh a x} a       | c = Primitive of $\operatorname{sech}^2 a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x \operatorname{sech}^2 a x \ \mathrm d x       | r = x \left({\frac {\tanh a x} a}\right) - \int \left({\frac {\tanh a x} a}\right) \times 1 \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = \frac {x \tanh a x} a - \frac 1 a \int \tanh a x \ \mathrm d x + C       | c = Linear Combination of Integrals }} {{eqn | r = \frac {x \tanh a x} a - \frac 1 a \frac {\ln \left\vert{\cosh a x}\right\vert} a + C       | c = Primitive of $\tanh a x$ }} {{eqn | r = \frac {x \tanh a x} a - \frac 1 {a^2} \ln \left\vert{\cosh a x}\right\vert + 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 = 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 = \csch^2 a x       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = -\frac {\coth a x} a       | c = Primitive of $\csch^2 a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x \csch^2 a x \rd x       | r = x \paren {-\frac {\coth a x} a} - \int \paren {-\frac {\coth a x} a} \times 1 \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {-x \coth a x} a + \frac 1 a \int \coth a x \rd x + C       | c = Linear Combination of Integrals }} {{eqn | r = \frac {-x \coth a x} a + \frac 1 a \frac {\ln \size {\sinh a x} } a + C       | c = Primitive of $\coth a x$ }} {{eqn | r = \frac {-x \coth a x} a + \frac 1 {a^2} \ln \size {\sinh a x} + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\mathrm d x} {\sinh^2 a x}       | r = \int \operatorname{csch}^2 a x \ \mathrm d x       | c = Definition of Hyperbolic Cosecant }} {{eqn | r = - \frac {\coth a x} a + C       | c = Primitive of $\operatorname{csch}^2 a x$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\mathrm d x} {\cosh^2 a x}       | r = \int \operatorname{sech}^2 a x \ \mathrm d x       | c = Definition of Hyperbolic Secant }} {{eqn | r = \frac {\tanh a x} a + C       | c = Primitive of $\operatorname{sech}^2 a x$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \sinh a x \sinh p x \rd x       | r = \int \paren {\frac {\map \cosh {a x + p x} - \map \cosh {a x - p x} } 2} \rd x       | c = Simpson's Formula for Hyperbolic Sine by Hyperbolic Sine }} {{eqn | r = \frac 1 2 \int \map \cosh {a + p} x \rd x - \frac 1 2 \int \map \cosh {a - p} x \rd x       | c = Linear Combination of Integrals }} {{eqn | r = \frac 1 2 \frac {\map \sinh {a + p} x} {a + p} - \frac 1 2 \frac {\map \sinh {a - p} x} {a - p} + C       | c = Primitive of $\cosh a x$ }} {{eqn | r = \frac {\map \sinh {a + p} x} {2 \paren {a + p} } - \frac {\map \sinh {a - p} x} {2 \paren {a - p} } + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \cosh a x \cosh p x \rd x       | r = \int \paren {\frac {\map \cosh {a x + p x} + \map \cosh {a x - p x} } 2} \rd x       | c = Simpson's Formula for Hyperbolic Cosine by Hyperbolic Cosine }} {{eqn | r = \frac 1 2 \int \map \cosh {a + p} x \rd x + \frac 1 2 \int \map \cosh {a - p} x \rd x       | c = Linear Combination of Integrals }} {{eqn | r = \frac 1 2 \frac {\map \sinh {a + p} x} {a + p} + \frac 1 2 \frac {\map \sinh {a - p} x} {a - p} + C       | c = Primitive of $\cosh a x$ }} {{eqn | r = \frac {\map \sinh {a + p} x} {2 \paren {a + p} } + \frac {\map \sinh {a - p} x} {2 \paren {a - p} } + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \sinh a x \sin p x \ \mathrm d x       | r = \int \left({\frac {e^{a x} - e^{- a x} } 2}\right) \sin p x \ \mathrm d x       | c = Definition of Hyperbolic Sine }} {{eqn | r = \frac 1 2 \int e^{a x} \sin p x \ \mathrm d x - \frac 1 2 \int e^{- a x} \sin p x \ \mathrm d x       | c = Linear Combination of Integrals }} {{eqn | r = \frac 1 2 \left({\frac {e^{a x} \left({a \sin p x - p \cos p x}\right)} {a^2 + p^2} }\right) - \frac 1 2 \left({\frac {e^{-a x} \left({-a \sin p x - p \cos p x}\right)} {a^2 + p^2} }\right) + C       | c = Primitive of $e^{a x} \sin b x$ }} {{eqn | r = \frac 1 {\left({a^2 + p^2}\right)} \left({a \left({\frac {e^{a x} + e^{-a x} } 2}\right) \sin p x - p \left({\frac {e^{a x} - e^{-a x} } 2}\right) \cos p x}\right) + C       | c = factoring }} {{eqn | r = \frac 1 {\left({a^2 + p^2}\right)} \left({a \cosh a x \sin p x - p \left({\frac {e^{a x} - e^{-a x} } 2}\right) \cos p x}\right) + C       | c = Definition of Hyperbolic Cosine }} {{eqn | r = \frac 1 {\left({a^2 + p^2}\right)} \left({a \cosh a x \sin p x - p \sinh a x \cos p x}\right) + C       | c = Definition of Hyperbolic Sine }} {{eqn | r = \frac {a \cosh a x \sin p x - p \sinh a x \cos p x} {a^2 + p^2} + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \sinh a x \cos p x \ \mathrm d x       | r = \int \left({\frac {e^{a x} - e^{- a x} } 2}\right) \cos p x \ \mathrm d x       | c = Definition of Hyperbolic Sine }} {{eqn | r = \frac 1 2 \int e^{a x} \cos p x \ \mathrm d x - \frac 1 2 \int e^{- a x} \cos p x \ \mathrm d x       | c = Linear Combination of Integrals }} {{eqn | r = \frac 1 2 \left({\frac {e^{a x} \left({a \cos p x + p \sin p x}\right)} {a^2 + p^2} }\right) - \frac 1 2 \left({\frac {e^{-a x} \left({-a \cos p x + p \sin p x}\right)} {a^2 + p^2} }\right) + C       | c = Primitive of $e^{a x} \cos b x$ }} {{eqn | r = \frac 1 {\left({a^2 + p^2}\right)} \left({a \left({\frac {e^{a x} + e^{-a x} } 2}\right) \cos p x + p \left({\frac {e^{a x} - e^{-a x} } 2}\right) \sin p x}\right) + C       | c = factoring }} {{eqn | r = \frac 1 {\left({a^2 + p^2}\right)} \left({a \cosh a x \cos p x + p \left({\frac {e^{a x} - e^{-a x} } 2}\right) \sin p x}\right) + C       | c = Definition of Hyperbolic Cosine }} {{eqn | r = \frac 1 {\left({a^2 + p^2}\right)} \left({a \cosh a x \cos p x + p \sinh a x \sin p x}\right) + C       | c = Definition of Hyperbolic Sine }} {{eqn | r = \frac {a \cosh a x \cos p x + p \sinh a x \sin p x} {a^2 + p^2} + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \cosh a x \sin p x \ \mathrm d x       | r = \int \left({\frac {e^{a x} + e^{- a x} } 2}\right) \sin p x \ \mathrm d x       | c = Definition of Hyperbolic Cosine }} {{eqn | r = \frac 1 2 \int e^{a x} \sin p x \ \mathrm d x + \frac 1 2 \int e^{- a x} \sin p x \ \mathrm d x       | c = Linear Combination of Integrals }} {{eqn | r = \frac 1 2 \left({\frac {e^{a x} \left({a \sin p x - p \cos p x}\right)} {a^2 + p^2} }\right) + \frac 1 2 \left({\frac {e^{-a x} \left({-a \sin p x - p \cos p x}\right)} {a^2 + p^2} }\right) + C       | c = Primitive of $e^{a x} \sin b x$ }} {{eqn | r = \frac 1 {\left({a^2 + p^2}\right)} \left({a \left({\frac {e^{a x} - e^{-a x} } 2}\right) \sin p x - p \left({\frac {e^{a x} + e^{-a x} } 2}\right) \cos p x}\right) + C       | c = factoring }} {{eqn | r = \frac 1 {\left({a^2 + p^2}\right)} \left({a \sinh a x \sin p x - p \left({\frac {e^{a x} + e^{-a x} } 2}\right) \cos p x}\right) + C       | c = Definition of Hyperbolic Sine }} {{eqn | r = \frac 1 {\left({a^2 + p^2}\right)} \left({a \sinh a x \sin p x - p \cosh a x \cos p x}\right) + C       | c = Definition of Hyperbolic Cosine }} {{eqn | r = \frac {a \sinh a x \sin p x - p \cosh a x \cos p x} {a^2 + p^2} + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \cosh a x \cos p x \ \mathrm d x       | r = \int \left({\frac {e^{a x} + e^{- a x} } 2}\right) \cos p x \ \mathrm d x       | c = Definition of Hyperbolic Cosine }} {{eqn | r = \frac 1 2 \int e^{a x} \cos p x \ \mathrm d x + \frac 1 2 \int e^{- a x} \cos p x \ \mathrm d x       | c = Linear Combination of Integrals }} {{eqn | r = \frac 1 2 \left({\frac {e^{a x} \left({a \cos p x + p \sin p x}\right)} {a^2 + p^2} }\right) + \frac 1 2 \left({\frac {e^{-a x} \left({-a \cos p x + p \sin p x}\right)} {a^2 + p^2} }\right) + C       | c = Primitive of $e^{a x} \cos b x$ }} {{eqn | r = \frac 1 {\left({a^2 + p^2}\right)} \left({a \left({\frac {e^{a x} - e^{-a x} } 2}\right) \cos p x + p \left({\frac {e^{a x} + e^{-a x} } 2}\right) \sin p x}\right) + C       | c = factoring }} {{eqn | r = \frac 1 {\left({a^2 + p^2}\right)} \left({a \sinh a x \cos p x + p \left({\frac {e^{a x} + e^{-a x} } 2}\right) \sin p x}\right) + C       | c = Definition of Hyperbolic Sine }} {{eqn | r = \frac 1 {\left({a^2 + p^2}\right)} \left({a \sinh a x \cos p x + p \cosh a x \sin p x}\right) + C       | c = Definition of Hyperbolic Cosine }} {{eqn | r = \frac {a \sinh a x \cos p x + p \cosh a x \sin p x} {a^2 + p^2} + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\d x} {p + q \sinh a x}       | r =  \frac 2 a \int \frac {\d u} {p u^2 + 2 q u - p}       | c = Hyperbolic Tangent Half-Angle Substitution: $u = \tanh \dfrac {a x} 2$ }} {{end-eqn}} The discriminant of $p u^2 + 2 q u - p$ is $4 \paren {p^2 + q^2}$. Thus: {{begin-eqn}} {{eqn | l = \int \frac {\d x} {p + q \sinh a x}       | r = \frac 2 a \paren {\dfrac 1 {\sqrt {4 \paren {p^2 + q^2} } } \ln \size {\dfrac {2 p x + 2 q - \sqrt {4 \paren {p^2 + q^2} } } {2 p x + 2 q + \sqrt {4 \paren {p^2 + q^2} } } } } + C       | c = Primitive of $\dfrac 1 {a x^2 + b x + c}$ }} {{eqn | r = \frac 1 {a \sqrt {p^2 + q^2} } \ln \size {\dfrac {p x + q - \sqrt {p^2 + q^2} } {p x + q + \sqrt {p^2 + q^2} } } + C       | c = simplifying }} {{eqn | r = \frac 1 {a \sqrt {p^2 + q^2} } \ln \size {\frac {p \tanh \dfrac {a x} 2 + q - \sqrt {p^2 + q^2} } {p \tanh \dfrac {a x} 2 + q + \sqrt {p^2 + q^2} } } + C       | c = substituting for $u$ }} {{eqn | r = \frac 1 {a \sqrt {p^2 + q^2} } \ln \size {\frac {p \dfrac {e^\paren {a x / 2} - e^\paren {-a x / 2} } {e^\paren {a x / 2} + e^\paren {-a x / 2} } + q - \sqrt {p^2 + q^2} } {p \dfrac {e^\paren {a x / 2} - e^\paren {-a x / 2} } {e^\paren {a x / 2} + e^\paren {-a x / 2} } + q + \sqrt {p^2 + q^2} } } + C       | c = {{Defof|Hyperbolic Tangent|index = 2}} }} {{eqn | r = \frac 1 {a \sqrt {p^2 + q^2} } \ln \size {\frac {p \dfrac {e^{a x} - 1} {e^{a x} + 1} + q - \sqrt {p^2 + q^2} } {p \dfrac {e^{a x} - 1} {e^{a x} + 1} + q + \sqrt {p^2 + q^2} } } + C       | c =  }} {{eqn | r = \frac 1 {a \sqrt {p^2 + q^2} } \ln \size {\frac {p \paren {e^{a x} - 1} + q \paren {e^{a x} + 1} - \sqrt {p^2 + q^2} \paren {e^{a x} + 1} } {p \paren {e^{a x} - 1} + q \paren {e^{a x} + 1} + \sqrt {p^2 + q^2} \paren {e^{a x} + 1} } } + C       | c =  }} {{end-eqn}} {{finish}}
{{begin-eqn}} {{eqn | l = \int \frac {\mathrm d x} {q + p \operatorname{sech} a x}       | r = \frac 1 q \int \frac {q \ \mathrm d x} {q + p \operatorname{sech} a x}       | c = multiplying top and bottom by $q$ }} {{eqn | r = \frac 1 q \int \frac {\left({q + p \operatorname{sech} a x - p \operatorname{sech} a x}\right) \ \mathrm d x} {q + p \operatorname{sech} a x}       | c =  }} {{eqn | r = \frac 1 q \int \frac {\left({q + p \operatorname{sech} a x}\right) \ \mathrm d x} {q + p \operatorname{sech} a x} - \frac p q \int \frac {\operatorname{sech} a x \ \mathrm d x} {q + p \operatorname{sech} a x}       | c = Linear Combination of Integrals }} {{eqn | r = \frac 1 q \int \mathrm d x - \frac p q \int \frac {\operatorname{sech} a x \ \mathrm d x} {q + p \operatorname{sech} a x}       | c = simplifying }} {{eqn | r = \frac x q - \frac p q \int \frac {\operatorname{sech} a x \ \mathrm d x} {q + p \operatorname{sech} a x} + C       | c = Primitive of Constant }} {{eqn | r = \frac x q - \frac p q \int \frac {\mathrm d x} {\frac q {\operatorname{sech} a x} + p} + C       | c = multiplying top and bottom by $\dfrac 1 {\operatorname{sech} a x}$ }} {{eqn | r = \frac x q - \frac p q \int \frac {\mathrm d x} {p + q \cosh a x} + C       | c = Definition of Hyperbolic Secant }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\d x} {q + p \csch a x}       | r = \frac 1 q \int \frac {q \rd x} {q + p \csch a x}       | c = multiplying top and bottom by $q$ }} {{eqn | r = \frac 1 q \int \frac {\paren {q + p \csch a x - p \csch a x} \rd x} {q + p \csch a x}       | c =  }} {{eqn | r = \frac 1 q \int \frac {\paren {q + p \csch a x} \rd x} {q + p \csch a x} - \frac p q \int \frac {\csch a x \rd x} {q + p \csch a x}       | c = Linear Combination of Integrals }} {{eqn | r = \frac 1 q \int \d x - \frac p q \int \frac {\csch a x \rd x} {q + p \csch a x}       | c = simplifying }} {{eqn | r = \frac x q - \frac p q \int \frac {\csch a x \rd x} {q + p \csch a x} + C       | c = Primitive of Constant }} {{eqn | r = \frac x q - \frac p q \int \frac {\d x} {\frac q {\csch a x} + p} + C       | c = multiplying top and bottom by $\dfrac 1 {\csch a x}$ }} {{eqn | r = \frac x q - \frac p q \int \frac {\d x} {p + q \sinh a x} + C       | c = {{Defof|Hyperbolic Cosecant|index = 2}} }} {{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^m       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u} {\mathrm d x}       | r = m x^{m - 1}       | 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^m \sinh a x \ \mathrm d x       | r = x^m \left({\frac {\cosh a x} a}\right) - \int \left({\frac {\cosh a x} a}\right) \left({m x^{m - 1} }\right) \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^m \cosh a x} a - \frac m a \int x^{m - 1} \cosh a 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: :$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = x^m       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = m x^{m - 1}       | c = Derivative of Power }} {{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 x^m \cosh a x \rd x       | r = x^m \paren {\frac {\sinh a x} a} - \int \paren {\frac {\sinh a x} a} \paren {m x^{m - 1} } \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^m \sinh a x} a - \frac m a \int x^{m - 1} \sinh a x \rd x + C       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} {{qed}}