What is the integral of cos(sqrt(x+3)) ?

The integral of cos(sqrt(x+3)) is 2(sin(sqrt(x+3))sqrt(x+3)+cos(sqrt(x+3)))+C

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integral of cos(sqrt(x+3))

\int cos (x + 3) d x

2 (sin (x + 3) x + 3 + cos (x + 3)) + C

Solution steps

\int cos (x + 3) d x

Apply u-substitution

: \int cos (u) d u

= \int cos (u) d u

Apply u-substitution

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

= \int cos (v) \cdot 2 v d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int cos (v) v d v

Apply Integration By Parts

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

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

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

= 2 (v sin (v) - (- cos (v)))

Substitute back

= 2 (x + 3 sin (x + 3) - (- cos (x + 3)))

Simplify

= 2 (sin (x + 3) x + 3 + cos (x + 3))

Add a constant to the solution

= 2 (sin (x + 3) x + 3 + cos (x + 3)) + C

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What is the integral of cos(sqrt(x+3)) ?
The integral of cos(sqrt(x+3)) is 2(sin(sqrt(x+3))sqrt(x+3)+cos(sqrt(x+3)))+C