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

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

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

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

- cos^{3} (x) + C

Solution steps

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

Take the constant out:

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

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

Simplify

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

Apply u-substitution

: \int - u^{2} d u

= 3 \cdot \int - u^{2} d u

Take the constant out:

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

= 3 (- \int u^{2} d u)

Apply the Power Rule

: \frac{u ^{3}}{3}

= 3 (- \frac{u ^{3}}{3})

Substitute back

u = cos (x)

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

Simplify

3 (- \frac{cos ^{3} ( x )}{3}) : - cos^{3} (x)

= - cos^{3} (x)

Add a constant to the solution

= - cos^{3} (x) + C

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