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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int 2 cos^{2} (u) d u

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

Take the constant out:

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

= 2 \cdot \int cos^{2} (u) d u

Rewrite using trig identities

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

Take the constant out:

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 1 d u = u

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

= 2 \cdot \frac{1}{2} (u + \frac{1}{2} sin (2 u))

Substitute back

u = x

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

Simplify

2 \cdot \frac{1}{2} (x + \frac{1}{2} sin (2 x)) : x + \frac{1}{2} sin (2 x)

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

Add a constant to the solution

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

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