What is the integral of sinh^4(x) ?

The integral of sinh^4(x) is 1/4 sinh^3(x)cosh(x)-3/8 (-x+1/2 sinh(2x))+C

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integral of sinh^4(x)

\int sinh^{4} (x) d x

\frac{1}{4} sinh^{3} (x) cosh (x) - \frac{3}{8} (- x + \frac{1}{2} sinh (2 x)) + C

Solution steps

\int sinh^{4} (x) d x

Apply Integral Reduction

= \frac{cosh ( x ) sinh ^{3} ( x )}{4} - \frac{3}{4} \cdot \int sinh^{2} (x) d x

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

= \frac{cosh ( x ) sinh ^{3} ( x )}{4} - \frac{3}{4} \cdot \frac{1}{2} (- x + \frac{1}{2} sinh (2 x))

Simplify

\frac{cosh ( x ) sinh ^{3} ( x )}{4} - \frac{3}{4} \cdot \frac{1}{2} (- x + \frac{1}{2} sinh (2 x)) : \frac{1}{4} sinh^{3} (x) cosh (x) - \frac{3}{8} (- x + \frac{1}{2} sinh (2 x))

= \frac{1}{4} sinh^{3} (x) cosh (x) - \frac{3}{8} (- x + \frac{1}{2} sinh (2 x))

Add a constant to the solution

= \frac{1}{4} sinh^{3} (x) cosh (x) - \frac{3}{8} (- x + \frac{1}{2} sinh (2 x)) + C

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e^{3 x}

\int (x + 3) (x^{2} - 15 x - 54) d x

What is the integral of sinh^4(x) ?
The integral of sinh^4(x) is 1/4 sinh^3(x)cosh(x)-3/8 (-x+1/2 sinh(2x))+C