What is the integral of (2x+3)/(sqrt(2x+1)) ?

The integral of (2x+3)/(sqrt(2x+1)) is 1/3 (2x+7)sqrt(2x+1)+C

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integral of (2x+3)/(sqrt(2x+1))

\int \frac{2 x + 3}{2 x + 1} d x

\frac{1}{3} (2 x + 7) 2 x + 1 + C

Solution steps

\int \frac{2 x + 3}{2 x + 1} d x

Apply u-substitution

: \int \frac{u + 2}{2 u} d u

= \int \frac{u + 2}{2 u} 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 \frac{u + 2}{u} d u

Expand

\frac{u + 2}{u} : u + \frac{2}{u}

= \frac{1}{2} \cdot \int u + \frac{2}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int u d u = \frac{2}{3} u^{\frac{3}{2}}

\int \frac{2}{u} d u = 4 u

= \frac{1}{2} (\frac{2}{3} u^{\frac{3}{2}} + 4 u)

Substitute back

u = 2 x + 1

= \frac{1}{2} (\frac{2}{3} (2 x + 1)^{\frac{3}{2}} + 4 2 x + 1)

Simplify

\frac{1}{2} (\frac{2}{3} (2 x + 1)^{\frac{3}{2}} + 4 2 x + 1) : \frac{1}{3} (2 x + 7) 2 x + 1

= \frac{1}{3} (2 x + 7) 2 x + 1

Add a constant to the solution

= \frac{1}{3} (2 x + 7) 2 x + 1 + C

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What is the integral of (2x+3)/(sqrt(2x+1)) ?
The integral of (2x+3)/(sqrt(2x+1)) is 1/3 (2x+7)sqrt(2x+1)+C