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

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 $u$

$\blacksquare$