What is the integral of 1/2 xsinh(2x) ?

The integral of 1/2 xsinh(2x) is 1/2 (1/2 xcosh(2x)-1/4 sinh(2x))+C

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integral of 1/2 xsinh(2x)

\int \frac{1}{2} x sinh (2 x) d x

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

Solution steps

\int \frac{1}{2} x sinh (2 x) d x

Take the constant out:

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

= \frac{1}{2} \cdot \int x sinh (2 x) d x

Apply Integration By Parts

: \frac{1}{2} x cosh (2 x) - \int \frac{1}{2} cosh (2 x) d x

= \frac{1}{2} (\frac{1}{2} x cosh (2 x) - \int \frac{1}{2} cosh (2 x) d x)

\int \frac{1}{2} cosh (2 x) d x = \frac{1}{4} sinh (2 x)

= \frac{1}{2} (\frac{1}{2} x cosh (2 x) - \frac{1}{4} sinh (2 x))

Add a constant to the solution

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

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What is the integral of 1/2 xsinh(2x) ?
The integral of 1/2 xsinh(2x) is 1/2 (1/2 xcosh(2x)-1/4 sinh(2x))+C