What is the integral of sin^4(x)cos^6(x) ?

The integral of sin^4(x)cos^6(x) is

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integral of sin^4(x)cos^6(x)

\int sin^{4} (x) cos^{6} (x) d x

\frac{1}{6} cos^{5} (x) sin (x) + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x))) - \frac{11}{10} (\frac{1}{8} cos^{7} (x) sin (x) + \frac{7}{8} (\frac{1}{6} cos^{5} (x) sin (x) + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x))))) + \frac{1}{10} cos^{9} (x) sin (x) + C

Solution steps

\int sin^{4} (x) cos^{6} (x) d x

Rewrite using trig identities

= \int (1 - cos^{2} (x))^{2} cos^{6} (x) d x

Expand

(1 - cos^{2} (x))^{2} cos^{6} (x) : cos^{6} (x) - 2 cos^{8} (x) + cos^{10} (x)

= \int cos^{6} (x) - 2 cos^{8} (x) + cos^{10} (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int cos^{6} (x) d x - \int 2 cos^{8} (x) d x + \int cos^{10} (x) d x

\int cos^{6} (x) d x = \frac{sin ( x ) cos ^{5} ( x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x)))

\int 2 cos^{8} (x) d x = 2 (\frac{sin ( x ) cos ^{7} ( x )}{8} + \frac{7}{8} (\frac{sin ( x ) cos ^{5} ( x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x)))))

\int cos^{10} (x) d x = \frac{sin ( x ) cos ^{9} ( x )}{10} + \frac{9}{10} (\frac{sin ( x ) cos ^{7} ( x )}{8} + \frac{7}{8} (\frac{sin ( x ) cos ^{5} ( x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x)))))

= \frac{sin ( x ) cos ^{5} ( x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x))) - 2 (\frac{sin ( x ) cos ^{7} ( x )}{8} + \frac{7}{8} (\frac{sin ( x ) cos ^{5} ( x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x))))) + \frac{sin ( x ) cos ^{9} ( x )}{10} + \frac{9}{10} (\frac{sin ( x ) cos ^{7} ( x )}{8} + \frac{7}{8} (\frac{sin ( x ) cos ^{5} ( x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x)))))

Simplify

\frac{sin ( x ) cos ^{5} ( x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x))) - 2 (\frac{sin ( x ) cos ^{7} ( x )}{8} + \frac{7}{8} (\frac{sin ( x ) cos ^{5} ( x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x))))) + \frac{sin ( x ) cos ^{9} ( x )}{10} + \frac{9}{10} (\frac{sin ( x ) cos ^{7} ( x )}{8} + \frac{7}{8} (\frac{sin ( x ) cos ^{5} ( x )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x))))) : \frac{1}{6} cos^{5} (x) sin (x) + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x))) - \frac{11}{10} (\frac{1}{8} cos^{7} (x) sin (x) + \frac{7}{8} (\frac{1}{6} cos^{5} (x) sin (x) + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x))))) + \frac{1}{10} cos^{9} (x) sin (x)

= \frac{1}{6} cos^{5} (x) sin (x) + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x))) - \frac{11}{10} (\frac{1}{8} cos^{7} (x) sin (x) + \frac{7}{8} (\frac{1}{6} cos^{5} (x) sin (x) + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x))))) + \frac{1}{10} cos^{9} (x) sin (x)

Add a constant to the solution

= \frac{1}{6} cos^{5} (x) sin (x) + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x))) - \frac{11}{10} (\frac{1}{8} cos^{7} (x) sin (x) + \frac{7}{8} (\frac{1}{6} cos^{5} (x) sin (x) + \frac{5}{6} (\frac{1}{4} cos^{3} (x) sin (x) + \frac{3}{8} (x + \frac{1}{2} sin (2 x))))) + \frac{1}{10} cos^{9} (x) sin (x) + C

y^{^{'}} = - \frac{4 t}{y}, y (0) = 2

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