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

The integral of (cos^2(x))/(sqrt(x)) is sqrt(x)+(sqrt(pi))/2 \C(sqrt(2)sqrt(2/pi)sqrt(x))+C

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integral of (cos^2(x))/(sqrt(x))

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

x + \frac{π}{2} C (2 \frac{2}{π} x) + C

Solution steps

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

Rewrite using trig identities

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

Expand

\frac{1 - sin ^{2} ( x )}{x} : \frac{1}{x} - \frac{sin ^{2} ( x )}{x}

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

\int \frac{1}{x} d x = 2 x

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

= 2 x - \frac{1}{2} (2 x - π C (2 \frac{2}{π} x))

Simplify

2 x - \frac{1}{2} (2 x - π C (2 \frac{2}{π} x)) : x + \frac{π}{2} C (2 \frac{2}{π} x)

= x + \frac{π}{2} C (2 \frac{2}{π} x)

Add a constant to the solution

= x + \frac{π}{2} C (2 \frac{2}{π} x) + C

x \to \infty lim (3^{x} - 1)

\frac{d}{d x} ((- x^{2} + 6 x - 4)^{5})

\frac{d}{d x} (\frac{4 - 3 x - x ^{2}}{x ^{2} - 6})

n = 0 \sum \infty \frac{1}{( n + 1 ) !}

f (x) = 2 x cos (x)

What is the integral of (cos^2(x))/(sqrt(x)) ?
The integral of (cos^2(x))/(sqrt(x)) is sqrt(x)+(sqrt(pi))/2 \C(sqrt(2)sqrt(2/pi)sqrt(x))+C