What is the integral of xsin(3x^2)cos^4(3x^2) ?

The integral of xsin(3x^2)cos^4(3x^2) is -1/30 cos^5(3x^2)+C

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integral of xsin(3x^2)cos^4(3x^2)

\int x sin (3 x^{2}) cos^{4} (3 x^{2}) d x

- \frac{1}{30} cos^{5} (3 x^{2}) + C

Solution steps

\int x sin (3 x^{2}) cos^{4} (3 x^{2}) d x

Apply u-substitution

: \int \frac{cos ^{4} ( 3 u ) sin ( 3 u )}{2} d u

= \int \frac{cos ^{4} ( 3 u ) sin ( 3 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^{4} (3 u) sin (3 u) d u

Apply u-substitution

: \int cos^{4} (v) sin (v) \frac{1}{3} d v

= \frac{1}{2} \cdot \int cos^{4} (v) sin (v) \frac{1}{3} 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}{3} \cdot \int cos^{4} (v) sin (v) d v

Apply u-substitution

: \int - w^{4} d w

= \frac{1}{2} \cdot \frac{1}{3} \cdot \int - w^{4} 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}{3} (- \int w^{4} d w)

Apply the Power Rule

: \frac{w ^{5}}{5}

= \frac{1}{2} \cdot \frac{1}{3} (- \frac{w ^{5}}{5})

Substitute back

= \frac{1}{2} \cdot \frac{1}{3} (- \frac{cos ^{5} ( 3 x ^{2} )}{5})

Simplify

\frac{1}{2} \cdot \frac{1}{3} (- \frac{cos ^{5} ( 3 x ^{2} )}{5}) : - \frac{1}{30} cos^{5} (3 x^{2})

= - \frac{1}{30} cos^{5} (3 x^{2})

Add a constant to the solution

= - \frac{1}{30} cos^{5} (3 x^{2}) + C

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What is the integral of xsin(3x^2)cos^4(3x^2) ?
The integral of xsin(3x^2)cos^4(3x^2) is -1/30 cos^5(3x^2)+C