What is the integral of (cos(5x))^2 ?

The integral of (cos(5x))^2 is 1/2 (x+1/10 sin(10x))+C

integral of (cos(5x))^2

\int (cos (5 x))^{2} d x

\frac{1}{2} (x + \frac{1}{10} sin (10 x)) + C

Solution steps

\int (cos (5 x))^{2} d x

Simplify

= \int cos^{2} (5 x) d x

Rewrite using trig identities

= \int \frac{1}{2} (1 + cos (10 x)) 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 (10 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 (10 x) d x)

\int 1 d x = x

\int cos (10 x) d x = \frac{1}{10} sin (10 x)

= \frac{1}{2} (x + \frac{1}{10} sin (10 x))

Add a constant to the solution

= \frac{1}{2} (x + \frac{1}{10} sin (10 x)) + C

t a y l or sin (\frac{π}{2} - x)

x \to - 1 - lim (x^{2})

x \to 3 - lim (\frac{x ^{2}}{x - 3})

\int e^{(a - s) x} d x

\int \frac{6 x ^{2} + 2 x - 1}{3 x ^{2}} d x

What is the integral of (cos(5x))^2 ?
The integral of (cos(5x))^2 is 1/2 (x+1/10 sin(10x))+C