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

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

Solution steps

\int sin (x + 1) d x

Apply u-substitution

= \int sin (u + 1) \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 + 1) u d u

Apply Integration By Parts

= 2 (- u cos (u + 1) - \int - cos (u + 1) d u)

\int - cos (u + 1) d u = - sin (u + 1)

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

Substitute back

u = x

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

Simplify

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

Add a constant to the solution

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

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