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

The integral of 5sin^3(5x)cos^3(5x) is -1/4 cos^4(5x)+1/6 cos^6(5x)+C

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

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

- \frac{1}{4} cos^{4} (5 x) + \frac{1}{6} cos^{6} (5 x) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int sin^{3} (u) cos^{3} (u) \frac{1}{5} d u

= 5 \cdot \int sin^{3} (u) cos^{3} (u) \frac{1}{5} d u

Take the constant out:

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

= 5 \cdot \frac{1}{5} \cdot \int sin^{3} (u) cos^{3} (u) d u

Rewrite using trig identities

= 5 \cdot \frac{1}{5} \cdot \int (1 - cos^{2} (u)) sin (u) cos^{3} (u) d u

Apply u-substitution

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

= 5 \cdot \frac{1}{5} \cdot \int - v^{3} (1 - v^{2}) d v

Expand

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

= 5 \cdot \frac{1}{5} \cdot \int - v^{3} + v^{5} 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

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

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

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

= 5 \cdot \frac{1}{5} (- \frac{v ^{4}}{4} + \frac{v ^{6}}{6})

Substitute back

= 5 \cdot \frac{1}{5} (- \frac{cos ^{4} ( 5 x )}{4} + \frac{cos ^{6} ( 5 x )}{6})

Simplify

5 \cdot \frac{1}{5} (- \frac{cos ^{4} ( 5 x )}{4} + \frac{cos ^{6} ( 5 x )}{6}) : - \frac{1}{4} cos^{4} (5 x) + \frac{1}{6} cos^{6} (5 x)

= - \frac{1}{4} cos^{4} (5 x) + \frac{1}{6} cos^{6} (5 x)

Add a constant to the solution

= - \frac{1}{4} cos^{4} (5 x) + \frac{1}{6} cos^{6} (5 x) + C

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What is the integral of 5sin^3(5x)cos^3(5x) ?
The integral of 5sin^3(5x)cos^3(5x) is -1/4 cos^4(5x)+1/6 cos^6(5x)+C