What is the integral of 3sin^5(x)cos^3(x) ?

The integral of 3sin^5(x)cos^3(x) is -3/4 cos^4(x)+cos^6(x)-3/8 cos^8(x)+C

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integral of 3sin^5(x)cos^3(x)

\int 3 sin^{5} (x) cos^{3} (x) d x

- \frac{3}{4} cos^{4} (x) + cos^{6} (x) - \frac{3}{8} cos^{8} (x) + C

Solution steps

\int 3 sin^{5} (x) cos^{3} (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^{5} (x) cos^{3} (x) d x

Rewrite using trig identities

= 3 \cdot \int (1 - cos^{2} (x))^{2} sin (x) cos^{3} (x) d x

Apply u-substitution

: \int - u^{3} (1 - u^{2})^{2} d u

= 3 \cdot \int - u^{3} (1 - u^{2})^{2} d u

Expand

- u^{3} (1 - u^{2})^{2} : - u^{3} + 2 u^{5} - u^{7}

= 3 \cdot \int - u^{3} + 2 u^{5} - u^{7} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 (- \int u^{3} d u + \int 2 u^{5} d u - \int u^{7} d u)

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

\int 2 u^{5} d u = \frac{u ^{6}}{3}

\int u^{7} d u = \frac{u ^{8}}{8}

= 3 (- \frac{u ^{4}}{4} + \frac{u ^{6}}{3} - \frac{u ^{8}}{8})

Substitute back

u = cos (x)

= 3 (- \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{3} - \frac{cos ^{8} ( x )}{8})

Simplify

3 (- \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{3} - \frac{cos ^{8} ( x )}{8}) : - \frac{3}{4} cos^{4} (x) + cos^{6} (x) - \frac{3}{8} cos^{8} (x)

= - \frac{3}{4} cos^{4} (x) + cos^{6} (x) - \frac{3}{8} cos^{8} (x)

Add a constant to the solution

= - \frac{3}{4} cos^{4} (x) + cos^{6} (x) - \frac{3}{8} cos^{8} (x) + C

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What is the integral of 3sin^5(x)cos^3(x) ?
The integral of 3sin^5(x)cos^3(x) is -3/4 cos^4(x)+cos^6(x)-3/8 cos^8(x)+C