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

The integral of (x+2)/(xsqrt(x+1)) is 2sqrt(x+1)-4(1/2 ln|sqrt(x+1)+1|-1/2 ln|sqrt(x+1)-1|)+C

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

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

2 x + 1 - 4 (\frac{1}{2} ln x + 1 + 1 - \frac{1}{2} ln x + 1 - 1) + C

Solution steps

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

Expand

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

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

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

\int \frac{2}{x x + 1} d x = - 4 (\frac{1}{2} ln x + 1 + 1 - \frac{1}{2} ln x + 1 - 1)

= 2 x + 1 - 4 (\frac{1}{2} ln x + 1 + 1 - \frac{1}{2} ln x + 1 - 1)

Add a constant to the solution

= 2 x + 1 - 4 (\frac{1}{2} ln x + 1 + 1 - \frac{1}{2} ln x + 1 - 1) + C

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What is the integral of (x+2)/(xsqrt(x+1)) ?
The integral of (x+2)/(xsqrt(x+1)) is 2sqrt(x+1)-4(1/2 ln|sqrt(x+1)+1|-1/2 ln|sqrt(x+1)-1|)+C