Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x

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Theorem

$\displaystyle \int \sinh a x \cosh a x \rd x = \frac {\sinh^2 a x} {2 a} + C$


Proof 1

\(\displaystyle \int \sinh a x \cosh a x \rd x\) \(=\) \(\displaystyle \int \frac {\sinh 2 a x} 2 \rd x\) Double Angle Formula for Hyperbolic Sine
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \int \sinh 2 a x \rd x\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \paren {\frac {\cosh 2 a x} {2 a} } + C\) Primitive of $\sinh a x$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \paren {\frac {1 + 2 \sinh^2 a x} {2 a} } + C\) Corollary to Double Angle Formula for Hyperbolic Cosine
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sinh^2 a x} {2 a} + \frac 1 {4 a} + C\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sinh^2 a x} {2 a} + C\) subsuming $\dfrac 1 {4 a}$ into arbitrary constant

$\blacksquare$


Proof 2

\(\displaystyle \int \sinh a x \cosh a x \rd x\) \(=\) \(\displaystyle \int \cosh a x \sinh a x \rd x\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\cosh^2 a x} {2 a} + C\) Primitive of $\cosh^n a x \sinh a x$ using $n = 1$
\(\displaystyle \) \(=\) \(\displaystyle \frac {1 + \sinh^2 a x} {2 a} + C\) Difference of Squares of Hyperbolic Cosine and Sine
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 a} + \frac {\sinh^2 a x} {2 a} + C\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sinh^2 a x} {2 a} + C\) subsuming $\dfrac 1 {2 a}$ into arbitrary constant

$\blacksquare$


Proof 3

\(\text {(1)}: \quad\) \(\displaystyle \int \sinh^n a x \cosh a x \rd x\) \(=\) \(\displaystyle \frac {\sinh^{n + 1} a x} {\paren {n + 1} a} + C\) Primitive of $\sinh^n a x \cosh a x$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \int \sinh a x \cosh a x \rd x\) \(=\) \(\displaystyle \frac {\sinh^2 a x} {2 a} + C\) setting $n = 1$ in $(1)$

$\blacksquare$


Proof 4

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:

\(\displaystyle u\) \(=\) \(\displaystyle \sinh a x\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \frac {\d u} {\d x}\) \(=\) \(\displaystyle a \cosh a x\) Derivative of $\sinh a x$


and let:

\(\displaystyle \frac {\d v} {\d x}\) \(=\) \(\displaystyle \cosh a x\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle v\) \(=\) \(\displaystyle \frac {\sinh a x} a\) Primitive of $\cosh a x$


Then:

\(\displaystyle \int \sinh a x \cosh a x \rd x\) \(=\) \(\displaystyle \paren {\sinh a x} \paren {\frac {\sinh a x} a} - \int \paren {\frac {\sinh a x} a} \paren {a \cosh a x} \rd x + C\) Integration by Parts
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sinh^2 a x} a - \int \sinh a x \cosh a x \rd x + C\) simplifying
\(\displaystyle \leadsto \ \ \) \(\displaystyle 2 \int \sinh a x \cosh a x \rd x\) \(=\) \(\displaystyle \frac {\sinh^2 a x} a + C\) gathering terms
\(\displaystyle \leadsto \ \ \) \(\displaystyle \int \sinh a x \cosh a x \rd x\) \(=\) \(\displaystyle \frac {\sinh^2 a x} {2 a} + C\) simplifying

$\blacksquare$


Proof 5

\(\displaystyle u\) \(=\) \(\displaystyle \sinh a x\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \frac {\d u} {\d x}\) \(=\) \(\displaystyle a \cosh a x\) Derivative of $\sinh a x$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \int \sinh a x \cosh a x \rd x\) \(=\) \(\displaystyle \int \frac u a \rd u\) Integration by Substitution
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 a \int u \rd u\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 a \frac {u^2} 2 + C\) Primitive of Power
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sinh^2 a x} {2 a} + C\) substituting for $u$

$\blacksquare$


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