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

The integral of (cos^2(x)sin(x))/2 is -1/6 cos^3(x)+C

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

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

- \frac{1}{6} cos^{3} (x) + C

Solution steps

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

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} (x) sin (x) d x

Apply u-substitution

: \int - u^{2} d u

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

Apply the Power Rule

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

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

Substitute back

u = cos (x)

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

Simplify

\frac{1}{2} (- \frac{cos ^{3} ( x )}{3}) : - \frac{1}{6} cos^{3} (x)

= - \frac{1}{6} cos^{3} (x)

Add a constant to the solution

= - \frac{1}{6} cos^{3} (x) + C

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What is the integral of (cos^2(x)sin(x))/2 ?
The integral of (cos^2(x)sin(x))/2 is -1/6 cos^3(x)+C