What is the integral of t^2cos^2(t^3)sin(t^3) ?

The integral of t^2cos^2(t^3)sin(t^3) is -1/9 cos^3(t^3)+C

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integral of t^2cos^2(t^3)sin(t^3)

\int t^{2} cos^{2} (t^{3}) sin (t^{3}) d t

- \frac{1}{9} cos^{3} (t^{3}) + C

Solution steps

\int t^{2} cos^{2} (t^{3}) sin (t^{3}) d t

Apply u-substitution

: \int \frac{u cos ^{2} ( u ^{\frac{3}{2}} ) sin ( u ^{\frac{3}{2}} )}{2 t} d u

= \int \frac{cos ^{2} ( u ) sin ( 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^{2} (u) sin (u) d u

Apply u-substitution

: \int - v^{2} d v

= \frac{1}{3} \cdot \int - v^{2} d v

Take the constant out:

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

= \frac{1}{3} (- \int v^{2} d v)

Apply the Power Rule

: \frac{v ^{3}}{3}

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

Substitute back

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

Simplify

\frac{1}{3} (- \frac{cos ^{3} ( t ^{3} )}{3}) : - \frac{1}{9} cos^{3} (t^{3})

= - \frac{1}{9} cos^{3} (t^{3})

Add a constant to the solution

= - \frac{1}{9} cos^{3} (t^{3}) + C

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What is the integral of t^2cos^2(t^3)sin(t^3) ?
The integral of t^2cos^2(t^3)sin(t^3) is -1/9 cos^3(t^3)+C