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

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

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integral of sin(sqrt(2x))

\int sin (2 x) d x

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

Solution steps

\int sin (2 x) d x

Apply u-substitution

: \int u sin (u) d u

= \int u sin (u) d u

Apply Integration By Parts

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

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

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

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

Substitute back

u = 2 x

= - 2 x cos (2 x) - (- sin (2 x))

Simplify

- 2 x cos (2 x) - (- sin (2 x)) : - 2 x cos (2 x) + sin (2 x)

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

Add a constant to the solution

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

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