What is the integral of cos^3(8-x)sin(8-x) ?

The integral of cos^3(8-x)sin(8-x) is 1/4 cos^4(8-x)+C

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integral of cos^3(8-x)sin(8-x)

\int cos^{3} (8 - x) sin (8 - x) d x

\frac{1}{4} cos^{4} (8 - x) + C

Solution steps

\int cos^{3} (8 - x) sin (8 - x) d x

Apply u-substitution

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

Take the constant out:

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

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

Apply u-substitution

= - \int - v^{3} d v

Take the constant out:

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

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

Apply the Power Rule

= - (- \frac{v ^{4}}{4})

Substitute back

= - (- \frac{cos ^{4} ( 8 - x )}{4})

Simplify

- (- \frac{cos ^{4} ( 8 - x )}{4}) : \frac{1}{4} cos^{4} (8 - x)

= \frac{1}{4} cos^{4} (8 - x)

Add a constant to the solution

= \frac{1}{4} cos^{4} (8 - x) + C

What is the integral of cos^3(8-x)sin(8-x) ?
The integral of cos^3(8-x)sin(8-x) is 1/4 cos^4(8-x)+C