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

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

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

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

x^{2} - 2 x + ln ∣ x + 1 ∣ + \frac{2 x ^{2}}{x + 1} - \frac{2}{x + 1} + ln ∣ x ∣ - 3 + C

Solution steps

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

Expand

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

\int \frac{2 x ^{3}}{x ^{2} + x} d x = x^{2} - 2 x + 2 ln ∣ x + 1 ∣ - 3

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

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

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

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

Simplify

x^{2} - 2 x + 2 ln ∣ x + 1 ∣ - 3 + 2 (\frac{x ^{2}}{x + 1} - ln ∣ x + 1 ∣ - \frac{1}{x + 1}) + 2 ln ∣ x + 1 ∣ + ln ∣ x ∣ - ln ∣ x + 1 ∣ : x^{2} - 2 x + ln ∣ x + 1 ∣ + \frac{2 x ^{2}}{x + 1} - \frac{2}{x + 1} + ln ∣ x ∣ - 3

= x^{2} - 2 x + ln ∣ x + 1 ∣ + \frac{2 x ^{2}}{x + 1} - \frac{2}{x + 1} + ln ∣ x ∣ - 3

Add a constant to the solution

= x^{2} - 2 x + ln ∣ x + 1 ∣ + \frac{2 x ^{2}}{x + 1} - \frac{2}{x + 1} + ln ∣ x ∣ - 3 + C

y = 4 x^{3} - 2 x

x \to 5 lim (x (5))

\frac{d}{d x} (\frac{- 2}{x})

e^{2 x} + 2 x e^{2 x}

\frac{d}{d x} (cos (\frac{1 - e ^{5 x}}{1 + e ^{5 x}}))

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