What is the integral of x^2sin(4x^3+7) ?

The integral of x^2sin(4x^3+7) is -1/12 cos(4x^3+7)+C

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integral of x^2sin(4x^3+7)

\int x^{2} sin (4 x^{3} + 7) d x

- \frac{1}{12} cos (4 x^{3} + 7) + C

Solution steps

\int x^{2} sin (4 x^{3} + 7) d x

Apply u-substitution

: \int \frac{u sin ( 4 u ^{\frac{3}{2}} + 7 )}{2 x} d u

= \int \frac{sin ( 4 u + 7 )}{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 (4 u + 7) d u

Apply u-substitution

: \int sin (v) \frac{1}{4} d v

= \frac{1}{3} \cdot \int sin (v) \frac{1}{4} 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}{4} \cdot \int sin (v) d v

Use the common integral:

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

= \frac{1}{3} \cdot \frac{1}{4} (- cos (v))

Substitute back

= \frac{1}{3} \cdot \frac{1}{4} (- cos (4 x^{3} + 7))

Simplify

\frac{1}{3} \cdot \frac{1}{4} (- cos (4 x^{3} + 7)) : - \frac{1}{12} cos (4 x^{3} + 7)

= - \frac{1}{12} cos (4 x^{3} + 7)

Add a constant to the solution

= - \frac{1}{12} cos (4 x^{3} + 7) + C

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What is the integral of x^2sin(4x^3+7) ?
The integral of x^2sin(4x^3+7) is -1/12 cos(4x^3+7)+C