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

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

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

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

sin (x^{3} + 2) + C

Solution steps

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

Take the constant out:

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

= 3 \cdot \int x^{2} cos (x^{3} + 2) d x

Apply u-substitution

= 3 \cdot \int \frac{cos ( u )}{3} d u

Take the constant out:

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

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

Use the common integral:

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

= 3 \cdot \frac{1}{3} sin (u)

Substitute back

u = x^{3} + 2

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

Simplify

3 \cdot \frac{1}{3} sin (x^{3} + 2) : sin (x^{3} + 2)

= sin (x^{3} + 2)

Add a constant to the solution

= sin (x^{3} + 2) + C

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