integral of cos^5(xsi)n^4x

Solution

\int cos^{5} (x s i) n^{4} x

\frac{n ^{4}}{225 s ^{2} i ^{2}} (225 s i x sin (s i x) - 150 s i x sin^{3} (s i x) + 45 s i x sin^{5} (s i x) + 120 cos (s i x) + 20 cos^{3} (s i x) + 9 cos^{5} (s i x)) + C

Solution steps

\int cos^{5} (x s i) n^{4} x d x

Take the constant out:

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

= n^{4} \cdot \int cos^{5} (s i x) x d x

Apply u-substitution

= n^{4} \cdot \int \frac{u cos ^{5} ( u )}{s ^{2} i ^{2}} d u

Take the constant out:

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

= n^{4} \frac{1}{s ^{2} i ^{2}} \cdot \int u cos^{5} (u) d u

Apply Integration By Parts

= n^{4} \frac{1}{s ^{2} i ^{2}} (u (sin (u) - \frac{2}{3} sin^{3} (u) + \frac{1}{5} sin^{5} (u)) - \int sin (u) - \frac{2}{3} sin^{3} (u) + \frac{1}{5} sin^{5} (u) d u)

\int sin (u) - \frac{2}{3} sin^{3} (u) + \frac{1}{5} sin^{5} (u) d u = - \frac{8}{15} cos (u) - \frac{2}{9} cos^{3} (u) + \frac{2}{15} cos^{3} (u) - \frac{1}{25} cos^{5} (u)

= n^{4} \frac{1}{s ^{2} i ^{2}} (u (sin (u) - \frac{2}{3} sin^{3} (u) + \frac{1}{5} sin^{5} (u)) - (- \frac{8}{15} cos (u) - \frac{2}{9} cos^{3} (u) + \frac{2}{15} cos^{3} (u) - \frac{1}{25} cos^{5} (u)))

Substitute back

u = s i x

= n^{4} \frac{1}{s ^{2} i ^{2}} (s i x (sin (s i x) - \frac{2}{3} sin^{3} (s i x) + \frac{1}{5} sin^{5} (s i x)) - (- \frac{8}{15} cos (s i x) - \frac{2}{9} cos^{3} (s i x) + \frac{2}{15} cos^{3} (s i x) - \frac{1}{25} cos^{5} (s i x)))

Simplify

n^{4} \frac{1}{s ^{2} i ^{2}} (s i x (sin (s i x) - \frac{2}{3} sin^{3} (s i x) + \frac{1}{5} sin^{5} (s i x)) - (- \frac{8}{15} cos (s i x) - \frac{2}{9} cos^{3} (s i x) + \frac{2}{15} cos^{3} (s i x) - \frac{1}{25} cos^{5} (s i x))) : \frac{n ^{4}}{225 s ^{2} i ^{2}} (225 s i x sin (s i x) - 150 s i x sin^{3} (s i x) + 45 s i x sin^{5} (s i x) + 120 cos (s i x) + 20 cos^{3} (s i x) + 9 cos^{5} (s i x))

= \frac{n ^{4}}{225 s ^{2} i ^{2}} (225 s i x sin (s i x) - 150 s i x sin^{3} (s i x) + 45 s i x sin^{5} (s i x) + 120 cos (s i x) + 20 cos^{3} (s i x) + 9 cos^{5} (s i x))

Add a constant to the solution

= \frac{n ^{4}}{225 s ^{2} i ^{2}} (225 s i x sin (s i x) - 150 s i x sin^{3} (s i x) + 45 s i x sin^{5} (s i x) + 120 cos (s i x) + 20 cos^{3} (s i x) + 9 cos^{5} (s i x)) + C