What is the integral of 24sin^7(3x)cos^3(3x) ?

The integral of 24sin^7(3x)cos^3(3x) is -2cos^4(3x)+4cos^6(3x)-3cos^8(3x)+4/5 cos^{10}(3x)+C

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integral of 24sin^7(3x)cos^3(3x)

\int 24 sin^{7} (3 x) cos^{3} (3 x) d x

- 2 cos^{4} (3 x) + 4 cos^{6} (3 x) - 3 cos^{8} (3 x) + \frac{4}{5} cos^{10} (3 x) + C

Solution steps

\int 24 sin^{7} (3 x) cos^{3} (3 x) d x

Take the constant out:

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

= 24 \cdot \int sin^{7} (3 x) cos^{3} (3 x) d x

Apply u-substitution

: \int sin^{7} (u) cos^{3} (u) \frac{1}{3} d u

= 24 \cdot \int sin^{7} (u) cos^{3} (u) \frac{1}{3} d u

Take the constant out:

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

= 24 \cdot \frac{1}{3} \cdot \int sin^{7} (u) cos^{3} (u) d u

Rewrite using trig identities

= 24 \cdot \frac{1}{3} \cdot \int (1 - cos^{2} (u))^{3} sin (u) cos^{3} (u) d u

Apply u-substitution

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

= 24 \cdot \frac{1}{3} \cdot \int - v^{3} (1 - v^{2})^{3} d v

Expand

- v^{3} (1 - v^{2})^{3} : - v^{3} + 3 v^{5} - 3 v^{7} + v^{9}

= 24 \cdot \frac{1}{3} \cdot \int - v^{3} + 3 v^{5} - 3 v^{7} + v^{9} d v

Apply the Sum Rule:

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

= 24 \cdot \frac{1}{3} (- \int v^{3} d v + \int 3 v^{5} d v - \int 3 v^{7} d v + \int v^{9} d v)

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

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

\int 3 v^{7} d v = \frac{3 v ^{8}}{8}

\int v^{9} d v = \frac{v ^{10}}{10}

= 24 \cdot \frac{1}{3} (- \frac{v ^{4}}{4} + \frac{v ^{6}}{2} - \frac{3 v ^{8}}{8} + \frac{v ^{10}}{10})

Substitute back

= 24 \cdot \frac{1}{3} (- \frac{cos ^{4} ( 3 x )}{4} + \frac{cos ^{6} ( 3 x )}{2} - \frac{3 cos ^{8} ( 3 x )}{8} + \frac{cos ^{10} ( 3 x )}{10})

Simplify

24 \cdot \frac{1}{3} (- \frac{cos ^{4} ( 3 x )}{4} + \frac{cos ^{6} ( 3 x )}{2} - \frac{3 cos ^{8} ( 3 x )}{8} + \frac{cos ^{10} ( 3 x )}{10}) : - 2 cos^{4} (3 x) + 4 cos^{6} (3 x) - 3 cos^{8} (3 x) + \frac{4}{5} cos^{10} (3 x)

= - 2 cos^{4} (3 x) + 4 cos^{6} (3 x) - 3 cos^{8} (3 x) + \frac{4}{5} cos^{10} (3 x)

Add a constant to the solution

= - 2 cos^{4} (3 x) + 4 cos^{6} (3 x) - 3 cos^{8} (3 x) + \frac{4}{5} cos^{10} (3 x) + C

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What is the integral of 24sin^7(3x)cos^3(3x) ?
The integral of 24sin^7(3x)cos^3(3x) is -2cos^4(3x)+4cos^6(3x)-3cos^8(3x)+4/5 cos^{10}(3x)+C