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

The integral of cos^2(x)-cos^4(x) is xcos^2(x)sin^2(x)-1/2 (-1/4 xcos(4x)+1/16 sin(4x))+C

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

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

x cos^{2} (x) sin^{2} (x) - \frac{1}{2} (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x)) + C

Solution steps

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

Apply Integration By Parts

: x cos^{2} (x) sin^{2} (x) - \int \frac{1}{2} x sin (4 x) d x

= x cos^{2} (x) sin^{2} (x) - \int \frac{1}{2} x sin (4 x) d x

\int \frac{1}{2} x sin (4 x) d x = \frac{1}{2} (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x))

= x cos^{2} (x) sin^{2} (x) - \frac{1}{2} (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x))

Add a constant to the solution

= x cos^{2} (x) sin^{2} (x) - \frac{1}{2} (- \frac{1}{4} x cos (4 x) + \frac{1}{16} sin (4 x)) + C

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What is the integral of cos^2(x)-cos^4(x) ?
The integral of cos^2(x)-cos^4(x) is xcos^2(x)sin^2(x)-1/2 (-1/4 xcos(4x)+1/16 sin(4x))+C