hcju/proofwikiqa huggingface dataset:
With a view to expressing the problem 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^{n - 1} a x       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = \paren {n - 1} a \sinh^{n - 2} a x \cosh a x       | c = Chain Rule for Derivatives, Derivative of $\sinh a x$, Derivative of Power }} {{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 {\cosh a x} a       | c = Primitive of $\sinh a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \sinh^n a x \rd x       | r = \int \sinh^{n - 1} a x \sinh a x \rd x       | c =  }} {{eqn | r = \sinh^{n - 1} a x \paren {\frac {\cosh a x} a} + C       | c = Integration by Parts }} {{eqn | o =        | ro= -       | r = \int \paren {\frac {\cosh a x} a} \paren {\paren {n - 1} a \sinh ^{n - 2} a x \cosh a x} \rd x + C       | c =  }} {{eqn | r = \frac {\sinh^{n - 1} a x \cosh a x} a       | c = simplifying }} {{eqn | o =        | ro= -       | r = \int \paren {n - 1} \sinh^{n - 2} a x \cosh^2 a x \rd x + C       | c =  }} {{eqn | r = \frac {\sinh^{n - 1} a x \cosh a x} a       | c = Difference of Squares of $\cosh$ and $\sinh$ }} {{eqn | o =        | ro= -       | r = \int \paren {n - 1} \sinh^{n - 2} a x \paren {1 + \sinh^2 a x} \rd x + C       | c =  }} {{eqn | r = \frac {\sinh^{n - 1} a x \cosh a x} a - \paren {n - 1} \int \sinh^{n - 2} a x \rd x + C       | c = Linear Combination of Integrals }} {{eqn | o =        | ro= -       | r = \paren {n - 1} \int \sinh^n a x \rd x + C       | c =  }} {{eqn | ll= \leadsto       | l = n \int \sinh^n a x \rd x       | r = \frac {\sinh^{n - 1} a x \cosh a x} a - \paren {n - 1} \int \sinh^{n - 2} a x \rd x + C       | c = rearranging }} {{eqn | ll= \leadsto       | l = \int \sinh^n a x \rd x       | r = \frac {\sinh^{n - 1} a x \cosh a x} {a n} - \frac {n - 1} n \int \sinh^{n - 2} a x \rd x + C       | c = dividing both sides by $n$ }} {{end-eqn}} {{qed}}
With a view to expressing the problem 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 = \cosh^{n - 1} a x       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = \paren {n - 1} a \cosh^{n - 2} a x \sinh a x       | c = Chain Rule for Derivatives, Derivative of $\cosh a x$, 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 \cosh^n a x \rd x       | r = \int \cosh^{n - 1} a x \cosh a x \rd x       | c =  }} {{eqn | r = \cosh^{n - 1} a x \paren {\frac {\sinh a x} a} + C       | c = Integration by Parts }} {{eqn | o =        | ro= -       | r = \int \paren {\frac {\sinh a x} a} \paren {\paren {n - 1} a \cosh^{n - 2} a x \sinh a x} \rd x + C       | c =  }} {{eqn | r = \frac {\cosh^{n - 1} a x \sinh a x} a       | c = simplifying }} {{eqn | o =        | ro= -       | r = \int \paren {n - 1} \cosh^{n - 2} a x \sinh^2 a x \rd x + C       | c =  }} {{eqn | r = \frac {\cosh^{n - 1} a x \sinh a x} a       | c = Difference of Squares of $\cosh$ and $\sinh$ }} {{eqn | o =        | ro= -       | r = \int \paren {n - 1} \cosh^{n - 2} a x \paren {\cosh^2 a x - 1} \rd x + C       | c =  }} {{eqn | r = \frac {\cosh^{n - 1} a x \sinh a x} a + \paren {n - 1} \int \cosh^{n - 2} a x \rd x + C       | c = Linear Combination of Integrals }} {{eqn | o =        | ro= -       | r = \paren {n - 1} \int \cosh^n a x \rd x + C       | c =  }} {{eqn | ll= \leadsto       | l = n \int \cosh^n a x \rd x       | r = \frac {\cosh^{n - 1} a x \sinh a x} a + \paren {n - 1} \int \cosh^{n - 2} a x \rd x + C       | c = rearranging }} {{eqn | ll= \leadsto       | l = \int \cosh^n a x \rd x       | r = \frac {\cosh^{n - 1} a x \sinh a x} {a n} + \frac {n - 1} n \int \cosh^{n - 2} a x \rd x + C       | c = dividing both sides by $n$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = z       | r = \tanh a x       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d z} {\mathrm d x}       | r = a \operatorname{sech}^2 a x       | c = Derivative of $\tanh a x$ }} {{eqn | ll= \implies       | l = \int \tanh^n a x \operatorname{sech}^2 a x \ \mathrm d x       | r = \int \frac 1 a z^n \ \mathrm d z       | c = Integration by Substitution }} {{eqn | r = \frac 1 a \frac {z^{n + 1} } {n + 1}       | c = Primitive of Power }} {{eqn | r = \frac {\tanh^{n + 1} a x} {\left({n + 1}\right) a} + C       | c = substituting for $z$ and simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\dfrac \d {\d x} } {\tanh x}       | r = \sech^2 x       | c = Derivative of $\tanh x$ }} {{eqn | ll= \leadsto       | l = \map {\dfrac \d {\d x} } {\tanh a x}       | r = a \sech^2 a x       | c = Derivative of Function of Constant Multiple }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\dfrac \d {\d x} } {\coth x}       | r = -\csch^2 x       | c = Derivative of $\coth x$ }} {{eqn | ll= \leadsto       | l = \map {\dfrac \d {\d x} } {\coth a x}       | r = -a \csch^2 a x       | c = Derivative of Function of Constant Multiple }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = z       | r = \coth a x       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d z} {\d x}       | r = -a \csch^2 a x       | c = Derivative of $\coth a x$ }} {{eqn | ll= \leadsto       | l = \int \coth^n a x \csch^2 a x \rd x       | r = \int \frac {-1} a z^n \rd z       | c = Integration by Substitution }} {{eqn | r = \frac {-1} a \frac {z^{n + 1} } {n + 1}       | c = Primitive of Power }} {{eqn | r = \frac {-\coth^{n + 1} a x} {\paren {n + 1} a} + C       | c = substituting for $z$ and simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \coth^n a x \rd x       | r = \int \coth^{n - 2} a x \coth^2 a x \rd x       | c =  }} {{eqn | r = \int \coth^{n - 2} a x \paren {1 + \csch^2 a x} \rd x       | c = Difference of Squares of Hyperbolic Cotangent and Cosecant }} {{eqn | r = \int \coth^{n - 2} a x \csch^2 a x \rd x + \int \coth^{n - 2} \rd x       | c = Linear Combination of Integrals }} {{eqn | r = \frac {-\coth^{n - 1} a x} {a \paren {n - 1} } + \int \coth^{n - 2} a x \rd x + C       | c = Primitive of $\coth^n a x \csch^2 a x$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\dfrac \d {\d x} } {\sech x}       | r = -\sech x \tanh x       | c = Derivative of $\sech x$ }} {{eqn | ll= \leadsto       | l = \map {\dfrac \d {\d x} } {\sech a x}       | r = -a \sech a x \tanh a x       | c = Derivative of Function of Constant Multiple }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \map {\dfrac \d {\d x} } {\csch x}       | r = -\csch x \coth x       | c = Derivative of $\csch x$ }} {{eqn | ll= \leadsto       | l = \map {\dfrac \d {\d x} } {\csch a x}       | r = -a \csch a x \coth a x       | c = Derivative of Function of Constant Multiple }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \sech^{n - 2} a x       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = -a \paren {n - 2} \sech^{n - 3} a x \sech a x \tanh a x       | c = Derivative of Power, Derivative of $\sech a x$, Chain Rule for Derivatives }} {{eqn | r = -a \paren {n - 2} \sech^{n - 2} a x \tanh a x       | c =  }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = \sech^2 a x       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {\tanh a x} a       | c = Primitive of $\sech^2 a x$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \sech^n a x \rd x       | r = \int \sech^{n - 2} a x \sech^2 a x \rd x       | c =  }} {{eqn | r = \sech^{n - 2} a x \paren {\frac {\tanh a x} a}       | c = Integration by Parts }} {{eqn | o =        | ro= -       | r = \int \paren {\frac {\tanh a x} a} \paren {-a \paren {n - 2} \sech^{n - 2} a x \tanh a x} \rd x       | c =  }} {{eqn | r = \frac {\sech^{n - 2} a x \tanh a x} a + \paren {n - 2} \int \tanh^2 a x \sech^{n - 2} a x \rd x       | c = simplifying }} {{eqn | r = \frac {\sech^{n - 2} a x \tanh a x} a + \paren {n - 2} \int \paren {1 - \sech^2 a x} \sech^{n - 2} a x \rd x       | c = Sum of $\sech^2$ and $\tanh^2$ }} {{eqn | r = \frac {\sech^{n - 2} a x \tanh a x} a - \paren {n - 2} \int \sech^n a x \rd x       | c = Linear Combination of Integrals }} {{eqn | o =        | ro= +       | r = \paren {n - 2} \int \sech^{n - 2} a x \rd x       | c =  }} {{eqn | l = \paren {n - 1} \int \sech^n a x \rd x       | r = \frac {\sech^{n - 2} a x \tanh a x} a + \paren {n - 2} \int \sech^{n - 2} a x \rd x       | c = gathering terms }} {{eqn | l = \int \sech^n a x \rd x       | r = \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 = dividing by $n - 1$ }} {{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 = \csch^{n - 2} a x       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = -a \paren {n - 2} \csch^{n - 3} a x \csch a x \coth a x       | c = Derivative of Power, Derivative of $\csch a x$, Chain Rule for Derivatives }} {{eqn | r = -a \paren {n - 2} \csch^{n - 2} a x \coth a x       | c =  }} {{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 \csch^n a x \rd x       | r = \int \csch^{n - 2} a x \csch^2 a x \rd x       | c =  }} {{eqn | r = \csch^{n - 2} a x \paren {\frac {-\coth a x} a}       | c = Integration by Parts }} {{eqn | o =        | ro= -       | r = \int \paren {\frac {-\coth a x} a} \paren {-a \paren {n - 2} \csch^{n - 2} a x \coth a x} \rd x       | c =  }} {{eqn | r = \frac {-\csch^{n - 2} a x \coth a x} a - \paren {n - 2} \int \coth^2 a x \csch^{n - 2} a x \rd x       | c = simplifying }} {{eqn | r = \frac {-\csch^{n - 2} a x \coth a x} a - \paren {n - 2} \int \paren {1 + \csch^2 a x} \csch^{n - 2} a x \rd x       | c = Difference of $\coth^2$ and $\csch^2$ }} {{eqn | r = \frac {-\csch^{n - 2} a x \coth a x} a - \paren {n - 2} \int \csch^n a x \rd x       | c = Linear Combination of Integrals }} {{eqn | o =        | ro= -       | r = \paren {n - 2} \int \csch^{n - 2} a x \rd x       | c =  }} {{eqn | l = \paren {n - 1} \int \csch^n a x \rd x       | r = \frac {-\csch^{n - 2} a x \coth a x} a - \paren {n - 2} \int \csch^{n - 2} a x \rd x       | c = gathering terms }} {{eqn | l = \int \csch^n a x \rd x       | r = \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 = dividing by $n - 1$ }} {{end-eqn}} {{qed}}
With a view to expressing the problem 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^n}       | c =  }} {{eqn | r = x^{-n}       | c =  }} {{eqn | ll= \implies       | l = v       | r = \frac {x^{-n + 1} } {- n + 1}       | c = Primitive of Power }} {{eqn | r = \frac {-1} {\left({n - 1}\right) x^{n - 1} }       | c = simplifying }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \frac {\sinh a x \ \mathrm d x} {x^n}       | r = \sinh a x \left({\frac {-1} {\left({n - 1}\right) x^{n - 1} } }\right) - \int \left({\frac {-1} {\left({n - 1}\right) x^{n - 1} } }\right) \left({a \cosh a x}\right) \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = \frac {-\sinh a x} {\left({n - 1}\right) x^{n - 1} } + \frac a {n - 1} \int \frac {\cosh a x \ \mathrm d x} {x^{n - 1} } + C       | c = simplifying }} {{end-eqn}} {{qed}}
With a view to expressing the problem 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 = \cosh a x       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = a \sinh a x       | c = Derivative of $\cosh a x$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = \frac 1 {x^n}       | c =  }} {{eqn | r = x^{-n}       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {x^{-n + 1} } {- n + 1}       | c = Primitive of Power }} {{eqn | r = \frac {-1} {\paren {n - 1} x^{n - 1} }       | c = simplifying }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \frac {\cosh a x \rd x} {x^n}       | r = \cosh a x \paren {\frac {-1} {\paren {n - 1} x^{n - 1} } } - \int \paren {\frac {-1} {\paren {n - 1} x^{n - 1} } } \paren {a \sinh a x} \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {-\cosh a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\sinh a x \rd x} {x^{n - 1} } + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = u       | r = \tanh \frac x 2       | c =  }} {{eqn | ll= \leadsto       | l = \int \frac {\d x} {1 + \cosh x}       | r = \int \frac {\dfrac {2 \rd u} {1 - u^2} } {\dfrac {1 + u^2} {1 - u^2} + 1}       | c = Hyperbolic Tangent Half-Angle Substitution }} {{eqn | r = \int \frac {2 \rd u} {1 + u^2 + \paren {1 - u^2} }       | c = multiplying top and bottom by $1 - u^2$ }} {{eqn | r = \int \rd u       | c = simplifying }} {{eqn | r = u + C       | c = Primitive of Constant }} {{eqn | r = \tanh \frac x 2 + C       | c = substituting for $u$ }} {{eqn | ll= \leadsto       | l = \int \frac {\d x} {1 + \cosh a x}       | r = \frac 1 a \tanh \frac {a x} 2 + C       | c = Primitive of Function of Constant Multiple }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = u       | r = \tanh \frac x 2       | c =  }} {{eqn | ll= \leadsto       | l = \int \frac {\d x} {\cosh x - 1}       | r = \int \frac {\dfrac {2 \rd u} {1 - u^2} } {\dfrac {1 + u^2} {1 - u^2} - 1}       | c = Hyperbolic Tangent Half-Angle Substitution }} {{eqn | r = \int \frac {2 \rd u} {1 + u^2 - \paren {1 - u^2} }       | c = multiplying top and bottom by $1 - u^2$ }} {{eqn | r = \int \frac {\d u} {u^2}       | c = simplifying }} {{eqn | r = \frac {-1} u + C       | c = Primitive of Power }} {{eqn | r = \frac {-1} {\tanh \dfrac x 2} + C       | c = substituting for $u$ }} {{eqn | r = -\coth \frac x 2 + C       | c = Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent }} {{eqn | ll= \leadsto       | l = \int \frac {\d x} {\cosh a x - 1}       | r = \frac {-1} a \coth \frac {a x} 2 + C       | c = Primitive of Function of Constant Multiple }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\mathrm d x} {\left({\cosh a x + 1}\right)^2}       | r = \int \left({\frac 1 2 \operatorname{sech}^2 \frac {a x} 2}\right)^2 \ \mathrm d x       | c = Reciprocal of One Plus Hyperbolic Cosine }} {{eqn | r = \frac 1 4 \int \operatorname{sech}^4 \frac {a x} 2 \ \mathrm d x       | c = simplifying }} {{eqn | r = \frac 1 4 \left({\frac{\operatorname{sech}^2 \dfrac {a x} 2 \tanh \dfrac {a x} 2} {\dfrac {3 a} 2} + \frac 2 3 \int \operatorname{sech}^2 \frac {a x} 2 \ \mathrm d x}\right) + C       | c = Primitive of $\operatorname{sech}^n a x$ }} {{eqn | r = \frac 1 {6 a} \operatorname{sech}^2 \frac {a x} 2 \tanh \dfrac {a x} 2 + \frac 1 6 \int \operatorname{sech}^2 \frac {a x} 2 \ \mathrm d x + C       | c = simplifying }} {{eqn | r = \frac 1 {6 a} \operatorname{sech}^2 \frac {a x} 2 \tanh \dfrac {a x} 2 + \frac 1 6 \left({\frac 2 a \tanh \frac {a x} 2}\right) + C       | c = Primitive of $\operatorname{sech}^2 a x$ }} {{eqn | r = \frac 1 {6 a} \left({1 - \tanh^2 \frac {a x} 2}\right) \tan \dfrac {a x} 2 + \frac 2 {6 a} \tanh \frac {a x} 2 + C       | c = Sum of Squares of Hyperbolic Secant and Tangent }} {{eqn | r = \frac 1 {2 a} \tanh \frac {a x} 2 - \frac 1 {6 a} \tanh^3 \frac {a x} 2 + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\d x} {\paren {\cosh a x - 1}^2}       | r = \int \paren {\frac 1 2 \csch^2 \frac {a x} 2}^2 \rd x       | c = Reciprocal of Hyperbolic Cosine Minus One }} {{eqn | r = \frac 1 4 \int \csch^4 \frac {a x} 2 \rd x       | c = simplifying }} {{eqn | r = \frac 1 4 \paren {\frac {-\csch^2 \dfrac {a x} 2 \coth \dfrac {a x} 2} {\dfrac {3 a} 2} - \frac 2 3 \int \csch^2 \frac {a x} 2 \rd x} + C       | c = Primitive of $\csch^n a x$ }} {{eqn | r = \frac {-1} {6 a} \csch^2 \frac {a x} 2 \coth \dfrac {a x} 2 - \frac 1 6 \int \csch^2 \frac {a x} 2 \rd x + C       | c = simplifying }} {{eqn | r = \frac {-1} {6 a} \csch^2 \frac {a x} 2 \coth \dfrac {a x} 2 - \frac 1 6 \paren {\frac {-2} a \coth \frac {a x} 2} + C       | c = Primitive of $\csch^2 a x$ }} {{eqn | r = \frac {-1} {6 a} \paren {\coth^2 \frac {a x} 2 - 1} \coth \dfrac {a x} 2 + \frac 2 {6 a} \coth \frac {a x} 2 + C       | c = Difference of Squares of Hyperbolic Cotangent and Cosecant }} {{eqn | r = \frac 1 {2 a} \coth \frac {a x} 2 - \frac 1 {6 a} \coth^3 \frac {a x} 2 + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \sinh p x \cosh q x \ \mathrm d x       | r = \int \left({\frac {\sinh \left({p x + q x}\right) + \sinh \left({p x - q x}\right)} 2}\right) \ \mathrm d x       | c = Simpson's Formula for Hyperbolic Sine by Hyperbolic Cosine }} {{eqn | r = \frac 1 2 \int \sinh \left({p + q}\right) x \ \mathrm d x + \frac 1 2 \int \sinh \left({p - q}\right) x \ \mathrm d x       | c = Linear Combination of Integrals }} {{eqn | r = \frac 1 2 \frac {\cosh \left({p + q}\right) x} {p + q} + \frac 1 2 \frac {\cosh \left({p - q}\right) x} {p - q} + C       | c = Primitive of $\sinh a x$ }} {{eqn | r = \frac {\cosh \left({p + q}\right) x} {2 \left({p + q}\right)} + \frac {\cosh \left({p - q}\right) x} {2 \left({p - q}\right)} + 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^n a x \cosh a x \rd x       | r = \int \frac {u^n} a \rd u       | c = Integration by Substitution }} {{eqn | r = \frac 1 a \int u^n \rd u       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 a \frac {u^{n + 1} } {n + 1} + C       | c = Primitive of Power }} {{eqn | r = \frac {\sinh^{n + 1} a x} {\paren {n + 1} a} + C       | c = substituting for $u$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = u       | r = \cosh a x       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = a \sinh a x       | c = Derivative of $\cosh a x$ }} {{eqn | ll= \leadsto       | l = \int \cosh^n a x \sinh a x \rd x       | r = \int \frac {u^n} a \rd u       | c = Integration by Substitution }} {{eqn | r = \frac 1 a \int u^n \rd u       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 a \frac {u^{n + 1} } {n + 1} + C       | c = Primitive of Power }} {{eqn | r = \frac {\cosh^{n + 1} a x} {\paren {n + 1} a} + C       | c = substituting for $u$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \sinh^2 a x \cosh^2 a x \rd x       | r = \int \paren {\sinh a x \cosh a x}^2 \rd x       | c =  }} {{eqn | r = \int \paren {\frac {\sinh 2 a x} 2}^2 \rd x       | c = Double Angle Formula for Hyperbolic Sine }} {{eqn | r = \frac 1 4 \int \sinh^2 2 a x \rd x       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 4 \paren {\frac {\sinh 2 \paren {2 a x} } {4 \paren {2 a} } - \frac x 2} + C       | c = Primitive of $\sinh^2 a x$ }} {{eqn | r = \frac {\sinh 4 a x} {32 a} - \frac x 8 + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\mathrm d x} {\sinh a x \cosh a x}       | r = \int \frac {\operatorname{sech} a x \ \mathrm d x} {\sinh a x}       | c = Definition of Hyperbolic Secant }} {{eqn | r = \int \frac {\operatorname{sech}^2 a x \ \mathrm d x} {\sinh a x \operatorname{sech} a x}       | c = multiplying top and bottom by $\operatorname{sech} a x$ }} {{eqn | r = \int \frac {\operatorname{sech}^2 a x \ \mathrm d x} {\frac {\sinh a x} {\cosh a x} }       | c = Definition of Hyperbolic Secant }} {{eqn | r = \int \frac {\operatorname{sech}^2 a x \ \mathrm d x} {\tanh a x}       | c = Definition of Hyperbolic Tangent }} {{eqn | r = \frac 1 a \ln \left\vert{\tanh a x}\right\vert + C       | c = Primitive of $\dfrac {\operatorname{sech}^2 a x} {\tanh a x}$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \frac {\mathrm d} {\mathrm d x} \tanh a x       | r = a \operatorname{sech}^2 a x       | c = Derivative of $\tanh a x$ }} {{eqn | ll= \implies       | l = \int \frac {a \operatorname{sech}^2 a x \ \mathrm d x} {\tanh a x}       | r = \ln \left\vert{\tanh a x}\right\vert + C       | c = Primitive of Function under its Derivative }} {{eqn | ll= \implies       | l = \int \frac {\operatorname{sech}^2 a x \ \mathrm d x} {\tanh a x}       | r = \frac 1 a \ln \left\vert{\tanh a x}\right\vert + C       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\d x} {\sinh^2 a x \cosh a x}       | r = \int \frac {\paren {\cosh^2 a x - \sinh^2 a x} \rd x} {\sinh^2 a x \cosh a x}       | c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn | r = \int \frac {\cosh^2 a x \rd x} {\sinh^2 a x \cosh a x} - \int \frac {\sinh^2 a x \rd x} {\sinh^2 a x \cosh a x}       | c = Linear Combination of Integrals }} {{eqn | r = \int \frac {\cosh a x \rd x} {\sinh^2 a x} - \int \frac {\d x} {\cosh a x}       | c = simplifying }} {{eqn | r = \int \frac {\cosh a x \rd x} {\sinh^2 a x} - \int \sech a x \rd x       | c = {{Defof|Hyperbolic Secant|index = 2}} }} {{eqn | r = \int \frac {\coth a x \rd x} {\sinh a x} - \int \sech a x \rd x       | c = {{Defof|Hyperbolic Cotangent|index = 2}} }} {{eqn | r = \int \csch a x \coth a x \rd x - \int \sech a x \rd x       | c = {{Defof|Hyperbolic Cosecant|index = 2}} }} {{eqn | r = \frac {-\csch a x} a - \int \sech a x \rd x       | c = Primitive of $\csch^n a x \coth a x$ for $n = 1$ }} {{eqn | r = \frac {-\csch a x} a - \frac 1 a \map \arctan {\sinh a x}       | c = Primitive of $\sech a x$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\mathrm d x} {\sinh^2 a x \cosh a x}       | r = \int \frac {\left({\cosh^2 a x - \sinh^2 a x}\right) \ \mathrm d x} {\sinh a x \cosh^2 a x}       | c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn | r = \int \frac {\cosh^2 a x \ \mathrm d x} {\sinh a x \cosh^2 a x} - \int \frac {\sinh^2 a x \ \mathrm d x} {\sinh a x \cosh^2 a x}       | c = Linear Combination of Integrals }} {{eqn | r = \int \frac {\mathrm d x} {\sinh a x} - \int \frac {\sinh a x \ \mathrm d x} {\cosh^2 a x}       | c = simplifying }} {{eqn | r = \int \operatorname{csch} a x \ \mathrm d x - \int \frac {\sinh a x \ \mathrm d x} {\cosh^2 a x}       | c = Definition of Hyperbolic Cosecant }} {{eqn | r = \int \operatorname{csch} a x \ \mathrm d x - \int \frac {\tanh a x \ \mathrm d x} {\cosh a x}       | c = Definition of Hyperbolic Tangent }} {{eqn | r = \int \operatorname{csch} a x \ \mathrm d x - \int \operatorname{sech} a x \tanh a x \ \mathrm d x       | c = Definition of Hyperbolic Secant }} {{eqn | r = \int \operatorname{csch} a x \ \mathrm d x - \frac {-\operatorname{sech} a x} a + C       | c = Primitive of $\operatorname{sech}^n a x \tanh a x$ for $n = 1$ }} {{eqn | r = \frac 1 a \ln \left\vert {\tanh \frac {a x} 2} \right\vert + \frac {\operatorname{sech} a x} a + C       | c = Primitive of $\operatorname{csch} a x$ }} {{end-eqn}} {{qed}}