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

The integral of x^2sin(x^3+3) is -1/3 cos(x^3+3)+C

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integral of x^2sin(x^3+3)

\int x^{2} sin (x^{3} + 3) d x

- \frac{1}{3} cos (x^{3} + 3) + C

Solution steps

\int x^{2} sin (x^{3} + 3) d x

Apply u-substitution

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

= \int \frac{sin ( u + 3 )}{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 sin (u + 3) d u

Apply u-substitution

: \int sin (v) d v

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

Use the common integral:

\int sin (v) d v = - cos (v)

= \frac{1}{3} (- cos (v))

Substitute back

= \frac{1}{3} (- cos (x^{3} + 3))

Simplify

= - \frac{1}{3} cos (x^{3} + 3)

Add a constant to the solution

= - \frac{1}{3} cos (x^{3} + 3) + C

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What is the integral of x^2sin(x^3+3) ?
The integral of x^2sin(x^3+3) is -1/3 cos(x^3+3)+C