What is the integral of (x^3+2x^2+x+1)/(x^2+2x) ?

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

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integral of (x^3+2x^2+x+1)/(x^2+2x)

\int \frac{x ^{3} + 2 x ^{2} + x + 1}{x ^{2} + 2 x} d x

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

Solution steps

\int \frac{x ^{3} + 2 x ^{2} + x + 1}{x ^{2} + 2 x} d x

Long division

\frac{x ^{3} + 2 x ^{2} + x + 1}{x ^{2} + 2 x} : x + \frac{x + 1}{x ^{2} + 2 x}

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

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

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

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

Add a constant to the solution

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

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