What is the integral of xcosh(2x^2) ?

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

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integral of xcosh(2x^2)

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

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

Solution steps

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

Apply u-substitution

: \int \frac{cosh ( 2 u )}{2} d u

= \int \frac{cosh ( 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 cosh (2 u) d u

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

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

Simplify

\frac{1}{2} \cdot \frac{1}{2} sinh (2 x^{2}) : \frac{1}{4} sinh (2 x^{2})

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

Add a constant to the solution

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

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