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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

Expand

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

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

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

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

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

Simplify

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

Add a constant to the solution

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

x \to - \infty lim (4 x^{2})

\frac{x ^{3} + sin ( x )}{7 x - cot ( x )}

\int 2 (e^{2 t} + 2) d t

y^{^{'}} + \frac{1}{3} y = e^{x} y^{4}

\frac{\partial}{\partial x} (7 cos (y e^{x y}) sin (x))

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