What is the integral of (cosh(x))^3 ?

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

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integral of (cosh(x))^3

\int (cosh (x))^{3} d x

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

Solution steps

\int (cosh (x))^{3} d x

Simplify

= \int cosh^{3} (x) d x

Rewrite using trig identities

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

Apply u-substitution

: \int 1 + u^{2} d u

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

\int 1 d u = u

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

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

Substitute back

u = sinh (x)

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (x^{\frac{1}{2}} + \frac{Y ( x )}{2})

f (x) = \frac{x ^{2} - 1}{x ^{2} + x + 1}

k = 2 \sum \infty \frac{2}{k ^{2} - 1}

y^{^{'}} + 2 y = e^{- t}, y (0) = \frac{3}{2}

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