What is the integral of cosh^5(x) ?

The integral of cosh^5(x) is sinh(x)+(2sinh^3(x))/3+(sinh^5(x))/5+C

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integral of cosh^5(x)

\int cosh^{5} (x) d x

sinh (x) + \frac{2 sinh ^{3} ( x )}{3} + \frac{sinh ^{5} ( x )}{5} + C

Solution steps

\int cosh^{5} (x) d x

Rewrite using trig identities

= \int (1 + sinh^{2} (x))^{2} cosh (x) d x

Apply u-substitution

= \int (1 + u^{2})^{2} d u

Expand

(1 + u^{2})^{2} : 1 + 2 u^{2} + u^{4}

= \int 1 + 2 u^{2} + u^{4} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d u + \int 2 u^{2} d u + \int u^{4} d u

\int 1 d u = u

\int 2 u^{2} d u = \frac{2 u ^{3}}{3}

\int u^{4} d u = \frac{u ^{5}}{5}

= u + \frac{2 u ^{3}}{3} + \frac{u ^{5}}{5}

Substitute back

u = sinh (x)

= sinh (x) + \frac{2 sinh ^{3} ( x )}{3} + \frac{sinh ^{5} ( x )}{5}

Add a constant to the solution

= sinh (x) + \frac{2 sinh ^{3} ( x )}{3} + \frac{sinh ^{5} ( x )}{5} + C

What is the integral of cosh^5(x) ?
The integral of cosh^5(x) is sinh(x)+(2sinh^3(x))/3+(sinh^5(x))/5+C