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

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

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

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

- \frac{x + 1}{x} - \frac{1}{2} ln x + 1 + 1 + \frac{1}{2} ln x + 1 - 1 + C

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

= - \frac{x + 1}{x} - \frac{1}{2} ln x + 1 + 1 + \frac{1}{2} ln x + 1 - 1

Add a constant to the solution

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

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