What is the integral of x^2cos(x^3-1) ?

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

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integral of x^2cos(x^3-1)

\int x^{2} cos (x^{3} - 1) d x

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

Solution steps

\int x^{2} cos (x^{3} - 1) d x

Factor

x^{3} - 1 : - (- x^{3} + 1)

= \int x^{2} cos (- (- x^{3} + 1)) d x

Simplify

x^{2} cos (- (- x^{3} + 1)) : x^{2} cos (1 - x^{3})

= \int x^{2} cos (1 - x^{3}) d x

Apply u-substitution

= \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

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

Use the common integral:

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

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

Substitute back

u = 1 - x^{3}

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

Add a constant to the solution

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

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