What is the integral of (x^2)/(x^2+2x+5) ?

The integral of (x^2)/(x^2+2x+5) is -ln|x^2+2x+5|-3/2 arctan(1/2 (x+1))+x+C

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integral of (x^2)/(x^2+2x+5)

\int \frac{x ^{2}}{x ^{2} + 2 x + 5} d x

- ln x^{2} + 2 x + 5 - \frac{3}{2} arctan (\frac{1}{2} (x + 1)) + x + C

Solution steps

\int \frac{x ^{2}}{x ^{2} + 2 x + 5} d x

\frac{x ^{2}}{x ^{2} + 2 x + 5} = \frac{- 2 x - 5}{x ^{2} + 2 x + 5} + 1

= \int \frac{- 2 x - 5}{x ^{2} + 2 x + 5} + 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 - 5}{x ^{2} + 2 x + 5} d x + \int 1 d x

\int \frac{- 2 x - 5}{x ^{2} + 2 x + 5} d x = - ln x^{2} + 2 x + 5 - \frac{3}{2} arctan (\frac{1}{2} (x + 1))

\int 1 d x = x

= - ln x^{2} + 2 x + 5 - \frac{3}{2} arctan (\frac{1}{2} (x + 1)) + x

Add a constant to the solution

= - ln x^{2} + 2 x + 5 - \frac{3}{2} arctan (\frac{1}{2} (x + 1)) + x + C

What is the integral of (x^2)/(x^2+2x+5) ?
The integral of (x^2)/(x^2+2x+5) is -ln|x^2+2x+5|-3/2 arctan(1/2 (x+1))+x+C