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

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

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integral of cos(sqrt(2x+1))

\int cos (2 x + 1) d x

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

Solution steps

\int cos (2 x + 1) d x

Apply u-substitution

: \int cos (u) \frac{1}{2} d u

= \int cos (u) \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int cos (u) d u

Apply u-substitution

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

= \frac{1}{2} \cdot \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

= \frac{1}{2} \cdot 2 \cdot \int cos (v) v d v

Apply Integration By Parts

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

= \frac{1}{2} \cdot 2 (v sin (v) - \int sin (v) d v)

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

= \frac{1}{2} \cdot 2 (v sin (v) - (- cos (v)))

Substitute back

= \frac{1}{2} \cdot 2 (2 x + 1 sin (2 x + 1) - (- cos (2 x + 1)))

Simplify

\frac{1}{2} \cdot 2 (2 x + 1 sin (2 x + 1) - (- cos (2 x + 1))) : sin (2 x + 1) 2 x + 1 + cos (2 x + 1)

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

Add a constant to the solution

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

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