What is the integral of cos(8x^3+7)x^2 ?

The integral of cos(8x^3+7)x^2 is 1/24 sin(8x^3+7)+C

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integral of cos(8x^3+7)x^2

\int cos (8 x^{3} + 7) x^{2} d x

\frac{1}{24} sin (8 x^{3} + 7) + C

Solution steps

\int cos (8 x^{3} + 7) x^{2} d x

Apply u-substitution

= \int \frac{cos ( 8 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 cos (8 u + 7) d u

Apply u-substitution

= \frac{1}{3} \cdot \int cos (v) \frac{1}{8} 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}{8} \cdot \int cos (v) d v

Use the common integral:

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

= \frac{1}{3} \cdot \frac{1}{8} sin (v)

Substitute back

= \frac{1}{3} \cdot \frac{1}{8} sin (8 x^{3} + 7)

Simplify

\frac{1}{3} \cdot \frac{1}{8} sin (8 x^{3} + 7) : \frac{1}{24} sin (8 x^{3} + 7)

= \frac{1}{24} sin (8 x^{3} + 7)

Add a constant to the solution

= \frac{1}{24} sin (8 x^{3} + 7) + C

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What is the integral of cos(8x^3+7)x^2 ?
The integral of cos(8x^3+7)x^2 is 1/24 sin(8x^3+7)+C