What is the integral of sinh^{15}(x) ?

The integral of sinh^{15}(x) is

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integral of sinh^{15}(x)

\int sinh^{15} (x) d x

- cosh (x) + \frac{7 cosh ^{3} ( x )}{3} - \frac{21 cosh ^{5} ( x )}{5} + 5 cosh^{7} (x) - \frac{35 cosh ^{9} ( x )}{9} + \frac{21 cosh ^{11} ( x )}{11} - \frac{7 cosh ^{13} ( x )}{13} + \frac{cosh ^{15} ( x )}{15} + C

Solution steps

\int sinh^{15} (x) d x

Rewrite using trig identities

= \int (- 1 + cosh^{2} (x))^{7} sinh (x) d x

Apply u-substitution

: \int (- 1 + u^{2})^{7} d u

= \int (- 1 + u^{2})^{7} d u

Expand

(- 1 + u^{2})^{7} : - 1 + 7 u^{2} - 21 u^{4} + 35 u^{6} - 35 u^{8} + 21 u^{10} - 7 u^{12} + u^{14}

= \int - 1 + 7 u^{2} - 21 u^{4} + 35 u^{6} - 35 u^{8} + 21 u^{10} - 7 u^{12} + u^{14} 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 7 u^{2} d u - \int 21 u^{4} d u + \int 35 u^{6} d u - \int 35 u^{8} d u + \int 21 u^{10} d u - \int 7 u^{12} d u + \int u^{14} d u

\int 1 d u = u

\int 7 u^{2} d u = \frac{7 u ^{3}}{3}

\int 21 u^{4} d u = \frac{21 u ^{5}}{5}

\int 35 u^{6} d u = 5 u^{7}

\int 35 u^{8} d u = \frac{35 u ^{9}}{9}

\int 21 u^{10} d u = \frac{21 u ^{11}}{11}

\int 7 u^{12} d u = \frac{7 u ^{13}}{13}

\int u^{14} d u = \frac{u ^{15}}{15}

= - u + \frac{7 u ^{3}}{3} - \frac{21 u ^{5}}{5} + 5 u^{7} - \frac{35 u ^{9}}{9} + \frac{21 u ^{11}}{11} - \frac{7 u ^{13}}{13} + \frac{u ^{15}}{15}

Substitute back

u = cosh (x)

= - cosh (x) + \frac{7 cosh ^{3} ( x )}{3} - \frac{21 cosh ^{5} ( x )}{5} + 5 cosh^{7} (x) - \frac{35 cosh ^{9} ( x )}{9} + \frac{21 cosh ^{11} ( x )}{11} - \frac{7 cosh ^{13} ( x )}{13} + \frac{cosh ^{15} ( x )}{15}

Add a constant to the solution

= - cosh (x) + \frac{7 cosh ^{3} ( x )}{3} - \frac{21 cosh ^{5} ( x )}{5} + 5 cosh^{7} (x) - \frac{35 cosh ^{9} ( x )}{9} + \frac{21 cosh ^{11} ( x )}{11} - \frac{7 cosh ^{13} ( x )}{13} + \frac{cosh ^{15} ( x )}{15} + C

f (x) = x^{2} + 4, at x = 3

\int x (2 x + 3)^{99} d x

x^{2} - 7 x

\int_{0}^{8} x 1 + x d x

\frac{1}{6 x - 1}