What is the integral of (x^3+3x^2+5x+4)/(x^2+3x+2) ?

The integral of (x^3+3x^2+5x+4)/(x^2+3x+2) is (x^2)/2+ln|x+1|+2ln|x+2|+C

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integral of (x^3+3x^2+5x+4)/(x^2+3x+2)

\int \frac{x ^{3} + 3 x ^{2} + 5 x + 4}{x ^{2} + 3 x + 2} d x

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

Solution steps

\int \frac{x ^{3} + 3 x ^{2} + 5 x + 4}{x ^{2} + 3 x + 2} d x

Take the partial fraction of

\frac{x ^{3} + 3 x ^{2} + 5 x + 4}{x ^{2} + 3 x + 2} : x + \frac{1}{x + 1} + \frac{2}{x + 2}

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

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

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

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

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

Add a constant to the solution

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

\int 90 x^{8} (1 - x) d x

\frac{( 10 )}{5 \cdot s ^{2} + 7 s}

\int \frac{4 y ^{2} - 7 y - 12}{y ( y + 2 ) ( y - 3 )} d y

c y^{- 7}

\int 3 e^{\frac{x}{3}} d x

What is the integral of (x^3+3x^2+5x+4)/(x^2+3x+2) ?
The integral of (x^3+3x^2+5x+4)/(x^2+3x+2) is (x^2)/2+ln|x+1|+2ln|x+2|+C