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

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

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

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

ln x^{2} + 1 + x + C

Solution steps

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

\frac{( x + 1 ) ^{2}}{x ^{2} + 1} = \frac{2 x}{x ^{2} + 1} + 1

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

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

\int 1 d x = x

= ln x^{2} + 1 + x

Add a constant to the solution

= ln x^{2} + 1 + x + C

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