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

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

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

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

\frac{x ^{2}}{2} - 3 x + 4 ln ∣ x + 1 ∣ + 3 ln ∣ x + 2 ∣ + C

Solution steps

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

Long division

\frac{x ^{3} + 5}{x ^{2} + 3 x + 2} : x - 3 + \frac{7 x + 11}{x ^{2} + 3 x + 2}

= \int x - 3 + \frac{7 x + 11}{x ^{2} + 3 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 3 d x + \int \frac{7 x + 11}{x ^{2} + 3 x + 2} d x

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

\int 3 d x = 3 x

\int \frac{7 x + 11}{x ^{2} + 3 x + 2} d x = 4 ln ∣ x + 1 ∣ + 3 ln ∣ x + 2 ∣

= \frac{x ^{2}}{2} - 3 x + 4 ln ∣ x + 1 ∣ + 3 ln ∣ x + 2 ∣

Add a constant to the solution

= \frac{x ^{2}}{2} - 3 x + 4 ln ∣ x + 1 ∣ + 3 ln ∣ x + 2 ∣ + C

\int - 7 cos^{2} (x) tan^{3} (x) d x

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

\int x ln (2 - x) d x

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

f (x) = 7 sin (x) + 3 cos (x)

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