integral of cos^4(xsi)n^3x

Solution

\int cos^{4} (x s i) n^{3} x d x

\frac{n ^{3}}{128 s ^{2} i ^{2}} (24 s^{2} i^{2} x^{2} + 128 s i x cos^{3} (s i x) sin (s i x) - 12 s i x sin (4 s i x) + 32 cos^{4} (s i x) - 3 cos (4 s i x)) + C

Solution steps

\int cos^{4} (x s i) n^{3} x d x

Take the constant out:

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

= n^{3} \cdot \int cos^{4} (s i x) x d x

Apply u-substitution

= n^{3} \cdot \int \frac{u cos ^{4} ( 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^{3} \frac{1}{s ^{2} i ^{2}} \cdot \int u cos^{4} (u) d u

Apply Integration By Parts

= n^{3} \frac{1}{s ^{2} i ^{2}} (u (cos^{3} (u) sin (u) + \frac{3}{8} (u - \frac{1}{4} sin (4 u))) - \int cos^{3} (u) sin (u) + \frac{3}{8} (u - \frac{1}{4} sin (4 u)) d u)

\int cos^{3} (u) sin (u) + \frac{3}{8} (u - \frac{1}{4} sin (4 u)) d u = - \frac{cos ^{4} ( u )}{4} + \frac{3}{8} (\frac{u ^{2}}{2} + \frac{1}{16} cos (4 u))

= n^{3} \frac{1}{s ^{2} i ^{2}} (u (cos^{3} (u) sin (u) + \frac{3}{8} (u - \frac{1}{4} sin (4 u))) - (- \frac{cos ^{4} ( u )}{4} + \frac{3}{8} (\frac{u ^{2}}{2} + \frac{1}{16} cos (4 u))))

Substitute back

u = s i x

= n^{3} \frac{1}{s ^{2} i ^{2}} (s i x (cos^{3} (s i x) sin (s i x) + \frac{3}{8} (s i x - \frac{1}{4} sin (4 s i x))) - (- \frac{cos ^{4} ( s i x )}{4} + \frac{3}{8} (\frac{( s i x ) ^{2}}{2} + \frac{1}{16} cos (4 s i x))))

Simplify

n^{3} \frac{1}{s ^{2} i ^{2}} (s i x (cos^{3} (s i x) sin (s i x) + \frac{3}{8} (s i x - \frac{1}{4} sin (4 s i x))) - (- \frac{cos ^{4} ( s i x )}{4} + \frac{3}{8} (\frac{( s i x ) ^{2}}{2} + \frac{1}{16} cos (4 s i x)))) : \frac{n ^{3}}{128 s ^{2} i ^{2}} (24 s^{2} i^{2} x^{2} + 128 s i x cos^{3} (s i x) sin (s i x) - 12 s i x sin (4 s i x) + 32 cos^{4} (s i x) - 3 cos (4 s i x))

= \frac{n ^{3}}{128 s ^{2} i ^{2}} (24 s^{2} i^{2} x^{2} + 128 s i x cos^{3} (s i x) sin (s i x) - 12 s i x sin (4 s i x) + 32 cos^{4} (s i x) - 3 cos (4 s i x))

Add a constant to the solution

= \frac{n ^{3}}{128 s ^{2} i ^{2}} (24 s^{2} i^{2} x^{2} + 128 s i x cos^{3} (s i x) sin (s i x) - 12 s i x sin (4 s i x) + 32 cos^{4} (s i x) - 3 cos (4 s i x)) + C