What is the integral of xcos^2(2x^2)sin(2x^2) ?

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

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integral of xcos^2(2x^2)sin(2x^2)

\int x cos^{2} (2 x^{2}) sin (2 x^{2}) d x

- \frac{1}{12} cos^{3} (2 x^{2}) + C

Solution steps

\int x cos^{2} (2 x^{2}) sin (2 x^{2}) d x

Apply u-substitution

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

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

Take the constant out:

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

= \frac{1}{2} \cdot \int cos^{2} (2 u) sin (2 u) d u

Apply u-substitution

: \int cos^{2} (v) sin (v) \frac{1}{2} d v

= \frac{1}{2} \cdot \int cos^{2} (v) sin (v) \frac{1}{2} d v

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int cos^{2} (v) sin (v) d v

Apply u-substitution

: \int - w^{2} d w

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

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{2} (- \int w^{2} d w)

Apply the Power Rule

: \frac{w ^{3}}{3}

= \frac{1}{2} \cdot \frac{1}{2} (- \frac{w ^{3}}{3})

Substitute back

= \frac{1}{2} \cdot \frac{1}{2} (- \frac{cos ^{3} ( 2 x ^{2} )}{3})

Simplify

\frac{1}{2} \cdot \frac{1}{2} (- \frac{cos ^{3} ( 2 x ^{2} )}{3}) : - \frac{1}{12} cos^{3} (2 x^{2})

= - \frac{1}{12} cos^{3} (2 x^{2})

Add a constant to the solution

= - \frac{1}{12} cos^{3} (2 x^{2}) + C

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