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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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