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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Simplify

\frac{1}{3} (x - 3) 2 x + 3 + 3 2 x + 3 : \frac{1}{3} x 2 x + 3 + 2 2 x + 3

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

Add a constant to the solution

= \frac{1}{3} x 2 x + 3 + 2 2 x + 3 + C

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