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

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

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

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

x 2 x + 1 + 2 x + 1 + C

Solution steps

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

Expand

\frac{3 x + 2}{2 x + 1} : \frac{3 x}{2 x + 1} + \frac{2}{2 x + 1}

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

Apply the Sum Rule:

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

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

\int \frac{3 x}{2 x + 1} d x = (x - 1) 2 x + 1

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

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

Simplify

(x - 1) 2 x + 1 + 2 2 x + 1 : x 2 x + 1 + 2 x + 1

= x 2 x + 1 + 2 x + 1

Add a constant to the solution

= x 2 x + 1 + 2 x + 1 + C

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