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

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

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

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

\frac{1}{2} (x + \frac{1}{2} sin (2 x)) + cos (x) + C

Solution steps

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

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

\int sin (x) d x = - cos (x)

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

Simplify

= \frac{1}{2} (x + \frac{1}{2} sin (2 x)) + cos (x)

Add a constant to the solution

= \frac{1}{2} (x + \frac{1}{2} sin (2 x)) + cos (x) + C

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\int x (x - 9) d x

\int sin (9 x) cos^{5} (9 x) d x

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