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

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

Popular >

integral of cos^2(sqrt(x))

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

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

Solution steps

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

Rewrite using trig identities

= \int 1 - sin^{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

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

\int 1 d x = x

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

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

Simplify

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

Add a constant to the solution

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

\int x arcsin (2 x) d x

\int \frac{1}{cos ^{2} ( x ) tan ( x ) - 1} d x

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

\int \frac{5 x ^{2} - 36 x + 48}{x ( x - 4 ) ^{2}} d x

\int e^{x} arccos (e^{x}) d x

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