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

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

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integral of sin(2sqrt(x))

\int sin (2 x) d x

- x cos (2 x) + \frac{1}{2} sin (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 cos (2 u) - \int - \frac{1}{2} cos (2 u) d u

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

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

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

Substitute back

u = x

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

Simplify

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

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

Add a constant to the solution

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

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