integral of 3picos^3(5pix)sin^7(5pix)

Solution

\int 3 π cos^{3} (5 π x) sin^{7} (5 π x) d x

\frac{3}{5} (- \frac{1}{4} cos^{4} (5 π x) + \frac{1}{2} cos^{6} (5 π x) - \frac{3}{8} cos^{8} (5 π x) + \frac{1}{10} cos^{10} (5 π x)) + C

Solution steps

\int 3 π cos^{3} (5 π x) sin^{7} (5 π x) d x

Take the constant out:

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

= 3 π \cdot \int cos^{3} (5 π x) sin^{7} (5 π x) d x

Apply u-substitution

= 3 π \cdot \int cos^{3} (u) sin^{7} (u) \frac{1}{5 π} d u

Take the constant out:

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

= 3 π \frac{1}{5 π} \cdot \int cos^{3} (u) sin^{7} (u) d u

Rewrite using trig identities

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

Apply u-substitution

= 3 π \frac{1}{5 π} \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}

= 3 π \frac{1}{5 π} \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

= 3 π \frac{1}{5 π} (- \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}

= 3 π \frac{1}{5 π} (- \frac{v ^{4}}{4} + \frac{v ^{6}}{2} - \frac{3 v ^{8}}{8} + \frac{v ^{10}}{10})

Substitute back

= 3 π \frac{1}{5 π} (- \frac{cos ^{4} ( 5 π x )}{4} + \frac{cos ^{6} ( 5 π x )}{2} - \frac{3 cos ^{8} ( 5 π x )}{8} + \frac{cos ^{10} ( 5 π x )}{10})

Simplify

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

= \frac{3}{5} (- \frac{1}{4} cos^{4} (5 π x) + \frac{1}{2} cos^{6} (5 π x) - \frac{3}{8} cos^{8} (5 π x) + \frac{1}{10} cos^{10} (5 π x))

Add a constant to the solution

= \frac{3}{5} (- \frac{1}{4} cos^{4} (5 π x) + \frac{1}{2} cos^{6} (5 π x) - \frac{3}{8} cos^{8} (5 π x) + \frac{1}{10} cos^{10} (5 π x)) + C