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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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