What is the integral of ((x^3+20x^2+8))/(x+20) ?

The integral of ((x^3+20x^2+8))/(x+20) is (x^3)/3+8ln|x+20|+C

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integral of ((x^3+20x^2+8))/(x+20)

\int \frac{( x ^{3} + 20 x ^{2} + 8 )}{x + 20} d x

\frac{x ^{3}}{3} + 8 ln ∣ x + 20 ∣ + C

Solution steps

\int \frac{x ^{3} + 20 x ^{2} + 8}{x + 20} d x

Long division

\frac{x ^{3} + 20 x ^{2} + 8}{x + 20} : x^{2} + \frac{8}{x + 20}

= \int x^{2} + \frac{8}{x + 20} 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^{2} d x + \int \frac{8}{x + 20} d x

\int x^{2} d x = \frac{x ^{3}}{3}

\int \frac{8}{x + 20} d x = 8 ln ∣ x + 20 ∣

= \frac{x ^{3}}{3} + 8 ln ∣ x + 20 ∣

Add a constant to the solution

= \frac{x ^{3}}{3} + 8 ln ∣ x + 20 ∣ + C

\int 7 e^{x^{2}} d x

\int x^{1} e^{x} d x

\int (6 x + 4) d x

6 y^{^{''}} + 3 y^{^{'}} = 0

f (x) = \frac{2}{3 x ^{4}}

What is the integral of ((x^3+20x^2+8))/(x+20) ?
The integral of ((x^3+20x^2+8))/(x+20) is (x^3)/3+8ln|x+20|+C