What is the integral of x^2sinh(2x^3) ?

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

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integral of x^2sinh(2x^3)

\int x^{2} sinh (2 x^{3}) d x

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

Solution steps

\int x^{2} sinh (2 x^{3}) d x

Apply u-substitution

: \int \frac{u sinh ( 2 u ^{\frac{3}{2}} )}{2 x} d u

= \int \frac{sinh ( 2 u )}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int sinh (2 u) d u

Apply u-substitution

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

= \frac{1}{3} \cdot \int sinh (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}{3} \cdot \frac{1}{2} \cdot \int sinh (v) d v

Use the common integral:

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

= \frac{1}{3} \cdot \frac{1}{2} cosh (v)

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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