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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

Simplify

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

Rewrite using trig identities

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

Expand

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

\int - 10 t^{2} + 9 d t

\int \frac{5}{64 - 81 t ^{2}} d t

\int \frac{- 1}{x ^{2} - 2 x - 3} d x

\int (\frac{7}{x ^{6}}) d x

\int x e^{x^{2} + y^{2}} d x

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