What is the integral of cos(7x)(11+sin(7x))^5 ?

The integral of cos(7x)(11+sin(7x))^5 is

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integral of cos(7x)(11+sin(7x))^5

\int cos (7 x) (11 + sin (7 x))^{5} d x

\frac{161051}{7} sin (7 x) - \frac{73205}{14} cos^{2} (7 x) + \frac{13310}{21} sin^{3} (7 x) + \frac{605}{14} sin^{4} (7 x) + \frac{11}{7} sin^{5} (7 x) + \frac{1}{42} sin^{6} (7 x) + C

Solution steps

\int cos (7 x) (11 + sin (7 x))^{5} d x

Expand

cos (7 x) (11 + sin (7 x))^{5} : 161051 cos (7 x) + 73205 cos (7 x) sin (7 x) + 13310 sin^{2} (7 x) cos (7 x) + 1210 sin^{3} (7 x) cos (7 x) + 55 sin^{4} (7 x) cos (7 x) + sin^{5} (7 x) cos (7 x)

= \int 161051 cos (7 x) + 73205 cos (7 x) sin (7 x) + 13310 sin^{2} (7 x) cos (7 x) + 1210 sin^{3} (7 x) cos (7 x) + 55 sin^{4} (7 x) cos (7 x) + sin^{5} (7 x) cos (7 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 161051 cos (7 x) d x + \int 73205 cos (7 x) sin (7 x) d x + \int 13310 sin^{2} (7 x) cos (7 x) d x + \int 1210 sin^{3} (7 x) cos (7 x) d x + \int 55 sin^{4} (7 x) cos (7 x) d x + \int sin^{5} (7 x) cos (7 x) d x

\int 161051 cos (7 x) d x = \frac{161051}{7} sin (7 x)

\int 73205 cos (7 x) sin (7 x) d x = - \frac{73205}{14} cos^{2} (7 x)

\int 13310 sin^{2} (7 x) cos (7 x) d x = \frac{13310}{21} sin^{3} (7 x)

\int 1210 sin^{3} (7 x) cos (7 x) d x = \frac{605}{14} sin^{4} (7 x)

\int 55 sin^{4} (7 x) cos (7 x) d x = \frac{11}{7} sin^{5} (7 x)

\int sin^{5} (7 x) cos (7 x) d x = \frac{1}{42} sin^{6} (7 x)

= \frac{161051}{7} sin (7 x) - \frac{73205}{14} cos^{2} (7 x) + \frac{13310}{21} sin^{3} (7 x) + \frac{605}{14} sin^{4} (7 x) + \frac{11}{7} sin^{5} (7 x) + \frac{1}{42} sin^{6} (7 x)

Add a constant to the solution

= \frac{161051}{7} sin (7 x) - \frac{73205}{14} cos^{2} (7 x) + \frac{13310}{21} sin^{3} (7 x) + \frac{605}{14} sin^{4} (7 x) + \frac{11}{7} sin^{5} (7 x) + \frac{1}{42} sin^{6} (7 x) + C

\int \frac{4 sin ( 3 y ) - 6}{cos ( 3 y )} d y

\int \frac{x ^{3} + 2 x + 5}{x ^{2} - 1} d x

\int arccoth (x) d x

\int x^{2} cos (4 x^{3} - 1) d x

What is the integral of cos(7x)(11+sin(7x))^5 ?
The integral of cos(7x)(11+sin(7x))^5 is