What is the integral of cos(x)cos^2(4x) ?

The integral of cos(x)cos^2(4x) is cos^2(4x)sin(x)+2(-1/9 sin(9x)+1/7 sin(7x))+C

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integral of cos(x)cos^2(4x)

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

cos^{2} (4 x) sin (x) + 2 (- \frac{1}{9} sin (9 x) + \frac{1}{7} sin (7 x)) + C

Solution steps

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

Apply Integration By Parts

= cos^{2} (4 x) sin (x) - \int - 4 sin (8 x) sin (x) d x

\int - 4 sin (8 x) sin (x) d x = - 2 (- \frac{1}{9} sin (9 x) + \frac{1}{7} sin (7 x))

= cos^{2} (4 x) sin (x) - (- 2 (- \frac{1}{9} sin (9 x) + \frac{1}{7} sin (7 x)))

Simplify

= cos^{2} (4 x) sin (x) + 2 (- \frac{1}{9} sin (9 x) + \frac{1}{7} sin (7 x))

Add a constant to the solution

= cos^{2} (4 x) sin (x) + 2 (- \frac{1}{9} sin (9 x) + \frac{1}{7} sin (7 x)) + C

What is the integral of cos(x)cos^2(4x) ?
The integral of cos(x)cos^2(4x) is cos^2(4x)sin(x)+2(-1/9 sin(9x)+1/7 sin(7x))+C