What is the integral of cosh(2x)sinh^2(2x) ?

The integral of cosh(2x)sinh^2(2x) is 1/6 sinh^3(2x)+C

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integral of cosh(2x)sinh^2(2x)

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

\frac{1}{6} sinh^{3} (2 x) + C

Solution steps

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

Apply u-substitution

= \int cosh (u) sinh^{2} (u) \frac{1}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int cosh (u) sinh^{2} (u) d u

Apply u-substitution

= \frac{1}{2} \cdot \int v^{2} d v

Apply the Power Rule

= \frac{1}{2} \cdot \frac{v ^{3}}{3}

Substitute back

= \frac{1}{2} \cdot \frac{sinh ^{3} ( 2 x )}{3}

Simplify

\frac{1}{2} \cdot \frac{sinh ^{3} ( 2 x )}{3} : \frac{1}{6} sinh^{3} (2 x)

= \frac{1}{6} sinh^{3} (2 x)

Add a constant to the solution

= \frac{1}{6} sinh^{3} (2 x) + C

What is the integral of cosh(2x)sinh^2(2x) ?
The integral of cosh(2x)sinh^2(2x) is 1/6 sinh^3(2x)+C