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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

Expand

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

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

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

\int (5 x^{2} + 1) e^{(\frac{5}{3} x^{3} + x)} d x

\int 5 x^{2} - 4 d x

\frac{d ^{2}}{d x ^{2}} (e^{\frac{- x ^{2}}{2}})

y = 7 x - 6 x, (1, 1)

y^{^{''}} + 9 y^{^{'}} - 10 y = 0