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

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

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integral of cosh^3(x)

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

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

Solution steps

\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

\int_{- 7}^{7} 49 - x^{2} d x

\frac{\partial}{\partial x} (x^{3} sec^{2} (x y))

\frac{\partial}{\partial x} (13 x + 6 x y^{3})

(y^{11} x) \frac{d y}{d x} = 1 + x

\frac{d}{d t} (6 sin (2 t))

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