What is the integral of sin(sqrt(y)) ?

The integral of sin(sqrt(y)) is 2(-sqrt(y)cos(sqrt(y))+sin(sqrt(y)))+C

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integral of sin(sqrt(y))

\int sin (y) d y

2 (- y cos (y) + sin (y)) + C

Solution steps

\int sin (y) d y

Apply u-substitution

: \int sin (u) \cdot 2 u d u

= \int sin (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 (u) u d u

Apply Integration By Parts

: - u cos (u) - \int - cos (u) d u

= 2 (- u cos (u) - \int - cos (u) d u)

\int - cos (u) d u = - sin (u)

= 2 (- u cos (u) - (- sin (u)))

Substitute back

u = y

= 2 (- y cos (y) - (- sin (y)))

Simplify

= 2 (- y cos (y) + sin (y))

Add a constant to the solution

= 2 (- y cos (y) + sin (y)) + C

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