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

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

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integral of cos^2(-x)

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

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

Solution steps

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

Simplify

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

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

Rewrite using trig identities

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int 1 + cos (2 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

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

\int 1 d x = x

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

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

Add a constant to the solution

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

\int \frac{( 1 - x )}{8 + 2 x - x ^{2}} d x

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\int \frac{1}{64 sin ^{2} ( x ) cos ( x )} d x

\int \frac{sin ( x )}{cos ( 2 x )} d x

\int \frac{x + 2}{x ^{3} + 9 x} d x

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