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

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

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integral of cos(sqrt(x+1))

\int cos (x + 1) d x

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

Solution steps

\int cos (x + 1) d x

Apply u-substitution

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = x + 1

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

Simplify

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

Add a constant to the solution

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

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