What is the integral of sin(sqrt(x+1)) ?

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

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integral of sin(sqrt(x+1))

\int sin (x + 1) d x

2 (- cos (x + 1) x + 1 + sin (x + 1)) + C

Solution steps

\int sin (x + 1) d x

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 = x + 1

= 2 (- x + 1 cos (x + 1) - (- sin (x + 1)))

Simplify

= 2 (- cos (x + 1) x + 1 + sin (x + 1))

Add a constant to the solution

= 2 (- cos (x + 1) x + 1 + sin (x + 1)) + C

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