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

The integral of xsinh(2-x^2) is -1/2 cosh(2-x^2)+C

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integral of xsinh(2-x^2)

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

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

Solution steps

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

Apply u-substitution

= \int \frac{sinh ( 2 - u )}{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 sinh (2 - u) d u

Apply u-substitution

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

Take the constant out:

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

= \frac{1}{2} (- \int sinh (v) d v)

Use the common integral:

\int sinh (v) d v = cosh (v)

= \frac{1}{2} (- cosh (v))

Substitute back

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

Simplify

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

Add a constant to the solution

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

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