What is the integral of sin^7(xco)s^5x ?

The integral of sin^7(xco)s^5x is

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integral of sin^7(xco)s^5x

\int sin^{7} (x co) s^{5} x d x

\frac{s ^{5}}{3675 c ^{2} o ^{2}} (- 3675 co x cos (co x) + 3675 co x cos^{3} (co x) - 2205 co x cos^{5} (co x) + 525 co x cos^{7} (co x) + 280 sin^{3} (co x) + 126 sin^{5} (co x) + 75 sin^{7} (co x) + 1680 sin (co x)) + C

Solution steps

\int sin^{7} (x co) s^{5} x d x

Take the constant out:

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

= s^{5} \cdot \int sin^{7} (co x) x d x

Apply u-substitution

: \int \frac{u sin ^{7} ( u )}{c ^{2} o ^{2}} d u

= s^{5} \cdot \int \frac{u sin ^{7} ( u )}{c ^{2} o ^{2}} d u

Take the constant out:

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

= s^{5} \frac{1}{c ^{2} o ^{2}} \cdot \int u sin^{7} (u) d u

Apply Integration By Parts

: u (- cos (u) + cos^{3} (u) - \frac{3}{5} cos^{5} (u) + \frac{1}{7} cos^{7} (u)) - \int - cos (u) + cos^{3} (u) - \frac{3}{5} cos^{5} (u) + \frac{1}{7} cos^{7} (u) d u

= s^{5} \frac{1}{c ^{2} o ^{2}} (u (- cos (u) + cos^{3} (u) - \frac{3}{5} cos^{5} (u) + \frac{1}{7} cos^{7} (u)) - \int - cos (u) + cos^{3} (u) - \frac{3}{5} cos^{5} (u) + \frac{1}{7} cos^{7} (u) d u)

\int - cos (u) + cos^{3} (u) - \frac{3}{5} cos^{5} (u) + \frac{1}{7} cos^{7} (u) d u = - \frac{16}{35} sin (u) + \frac{9}{35} sin^{3} (u) - \frac{3}{25} sin^{5} (u) - \frac{1}{3} sin^{3} (u) + \frac{3}{35} sin^{5} (u) - \frac{1}{49} sin^{7} (u)

= s^{5} \frac{1}{c ^{2} o ^{2}} (u (- cos (u) + cos^{3} (u) - \frac{3}{5} cos^{5} (u) + \frac{1}{7} cos^{7} (u)) - (- \frac{16}{35} sin (u) + \frac{9}{35} sin^{3} (u) - \frac{3}{25} sin^{5} (u) - \frac{1}{3} sin^{3} (u) + \frac{3}{35} sin^{5} (u) - \frac{1}{49} sin^{7} (u)))

Substitute back

u = co x

= s^{5} \frac{1}{c ^{2} o ^{2}} (co x (- cos (co x) + cos^{3} (co x) - \frac{3}{5} cos^{5} (co x) + \frac{1}{7} cos^{7} (co x)) - (- \frac{16}{35} sin (co x) + \frac{9}{35} sin^{3} (co x) - \frac{3}{25} sin^{5} (co x) - \frac{1}{3} sin^{3} (co x) + \frac{3}{35} sin^{5} (co x) - \frac{1}{49} sin^{7} (co x)))

Simplify

s^{5} \frac{1}{c ^{2} o ^{2}} (co x (- cos (co x) + cos^{3} (co x) - \frac{3}{5} cos^{5} (co x) + \frac{1}{7} cos^{7} (co x)) - (- \frac{16}{35} sin (co x) + \frac{9}{35} sin^{3} (co x) - \frac{3}{25} sin^{5} (co x) - \frac{1}{3} sin^{3} (co x) + \frac{3}{35} sin^{5} (co x) - \frac{1}{49} sin^{7} (co x))) : \frac{s ^{5}}{3675 c ^{2} o ^{2}} (- 3675 co x cos (co x) + 3675 co x cos^{3} (co x) - 2205 co x cos^{5} (co x) + 525 co x cos^{7} (co x) + 280 sin^{3} (co x) + 126 sin^{5} (co x) + 75 sin^{7} (co x) + 1680 sin (co x))

= \frac{s ^{5}}{3675 c ^{2} o ^{2}} (- 3675 co x cos (co x) + 3675 co x cos^{3} (co x) - 2205 co x cos^{5} (co x) + 525 co x cos^{7} (co x) + 280 sin^{3} (co x) + 126 sin^{5} (co x) + 75 sin^{7} (co x) + 1680 sin (co x))

Add a constant to the solution

= \frac{s ^{5}}{3675 c ^{2} o ^{2}} (- 3675 co x cos (co x) + 3675 co x cos^{3} (co x) - 2205 co x cos^{5} (co x) + 525 co x cos^{7} (co x) + 280 sin^{3} (co x) + 126 sin^{5} (co x) + 75 sin^{7} (co x) + 1680 sin (co x)) + C

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