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

The integral of 3sin^4(x)cos(x) is 3/5 sin^5(x)+C

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

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

\frac{3}{5} sin^{5} (x) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int u^{4} d u

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

Apply the Power Rule

: \frac{u ^{5}}{5}

= 3 \cdot \frac{u ^{5}}{5}

Substitute back

u = sin (x)

= 3 \cdot \frac{sin ^{5} ( x )}{5}

Simplify

3 \cdot \frac{sin ^{5} ( x )}{5} : \frac{3}{5} sin^{5} (x)

= \frac{3}{5} sin^{5} (x)

Add a constant to the solution

= \frac{3}{5} sin^{5} (x) + C

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What is the integral of 3sin^4(x)cos(x) ?
The integral of 3sin^4(x)cos(x) is 3/5 sin^5(x)+C