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

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

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integral of sin^2(sqrt(x))

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

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

Solution steps

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

Apply u-substitution

: \int sin^{2} (u) \cdot 2 u d u

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

Apply Integration By Parts

: \frac{1}{2} u (u - \frac{1}{2} sin (2 u)) - \int \frac{1}{2} (u - \frac{1}{2} sin (2 u)) d u

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

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

= 2 (\frac{1}{2} u (u - \frac{1}{2} sin (2 u)) - \frac{1}{2} (\frac{u ^{2}}{2} + \frac{1}{4} cos (2 u)))

Substitute back

u = x

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

Simplify

2 (\frac{1}{2} x (x - \frac{1}{2} sin (2 x)) - \frac{1}{2} (\frac{( x ) ^{2}}{2} + \frac{1}{4} cos (2 x))) : x - \frac{1}{2} x sin (2 x) - \frac{x}{2} - \frac{1}{4} cos (2 x)

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

Add a constant to the solution

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

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