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

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

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integral of 4cos^3(x)sin(x)

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

- cos^{4} (x) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int - u^{3} d u

= 4 \cdot \int - u^{3} d u

Take the constant out:

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

= 4 (- \int u^{3} d u)

Apply the Power Rule

: \frac{u ^{4}}{4}

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

Substitute back

u = cos (x)

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

Simplify

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

= - cos^{4} (x)

Add a constant to the solution

= - cos^{4} (x) + C

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