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

The integral of (x^4+x^3)/(x-3) is (x^4)/4+x^3+(x^3)/3+(9x^2)/2+(3x^2)/2+36x+108ln|x-3|+C

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integral of (x^4+x^3)/(x-3)

\int \frac{x ^{4} + x ^{3}}{x - 3} d x

\frac{x ^{4}}{4} + x^{3} + \frac{x ^{3}}{3} + \frac{9 x ^{2}}{2} + \frac{3 x ^{2}}{2} + 36 x + 108 ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{x ^{4} + x ^{3}}{x - 3} d x

Expand

\frac{x ^{4} + x ^{3}}{x - 3} : \frac{x ^{4}}{x - 3} + \frac{x ^{3}}{x - 3}

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{x ^{4}}{x - 3} d x + \int \frac{x ^{3}}{x - 3} d x

\int \frac{x ^{4}}{x - 3} d x = \frac{x ^{4}}{4} + x^{3} + \frac{9 x ^{2}}{2} + 27 x + 81 ln ∣ x - 3 ∣

\int \frac{x ^{3}}{x - 3} d x = \frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} + 9 x + 27 ln ∣ x - 3 ∣

= \frac{x ^{4}}{4} + x^{3} + \frac{9 x ^{2}}{2} + 27 x + 81 ln ∣ x - 3 ∣ + \frac{x ^{3}}{3} + \frac{3 x ^{2}}{2} + 9 x + 27 ln ∣ x - 3 ∣

Simplify

= \frac{x ^{4}}{4} + x^{3} + \frac{x ^{3}}{3} + \frac{9 x ^{2}}{2} + \frac{3 x ^{2}}{2} + 36 x + 108 ln ∣ x - 3 ∣

Add a constant to the solution

= \frac{x ^{4}}{4} + x^{3} + \frac{x ^{3}}{3} + \frac{9 x ^{2}}{2} + \frac{3 x ^{2}}{2} + 36 x + 108 ln ∣ x - 3 ∣ + C

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What is the integral of (x^4+x^3)/(x-3) ?
The integral of (x^4+x^3)/(x-3) is (x^4)/4+x^3+(x^3)/3+(9x^2)/2+(3x^2)/2+36x+108ln|x-3|+C