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

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

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integral of 2xsinh(2x)

\int 2 x sinh (2 x) d x

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

Solution steps

\int 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

= 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

= 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)

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

Simplify

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

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

Add a constant to the solution

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

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