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

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

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

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

x^{2} + ln ∣ x ∣ + C

Solution steps

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

Simplify

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

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

Expand

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

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

= x^{2} + ln ∣ x ∣

Add a constant to the solution

= x^{2} + ln ∣ x ∣ + C

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