What is the integral of (2x^4-6x^3+5x)/(x+2) ?

The integral of (2x^4-6x^3+5x)/(x+2) is 42x^2+37x+70ln|x+2|+1/2 (x+2)^4-22/3 (x+2)^3-94+C

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integral of (2x^4-6x^3+5x)/(x+2)

\int \frac{2 x ^{4} - 6 x ^{3} + 5 x}{x + 2} d x

42 x^{2} + 37 x + 70 ln ∣ x + 2 ∣ + \frac{1}{2} (x + 2)^{4} - \frac{22}{3} (x + 2)^{3} - 94 + C

Solution steps

\int \frac{2 x ^{4} - 6 x ^{3} + 5 x}{x + 2} d x

Expand

\frac{2 x ^{4} - 6 x ^{3} + 5 x}{x + 2} : \frac{2 x ^{4}}{x + 2} - \frac{6 x ^{3}}{x + 2} + \frac{5 x}{x + 2}

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 ^{4}}{x + 2} d x - \int \frac{6 x ^{3}}{x + 2} d x + \int \frac{5 x}{x + 2} d x

\int \frac{2 x ^{4}}{x + 2} d x = 2 (12 x^{2} + 16 x + 16 ln ∣ x + 2 ∣ + \frac{1}{4} (x + 2)^{4} - \frac{8}{3} (x + 2)^{3} - 16)

\int \frac{6 x ^{3}}{x + 2} d x = 6 (- 3 x^{2} + \frac{1}{3} (x + 2)^{3} - 8 ln ∣ x + 2 ∣ + 12)

\int \frac{5 x}{x + 2} d x = 5 (x + 2 - 2 ln ∣ x + 2 ∣)

= 2 (12 x^{2} + 16 x + 16 ln ∣ x + 2 ∣ + \frac{1}{4} (x + 2)^{4} - \frac{8}{3} (x + 2)^{3} - 16) - 6 (- 3 x^{2} + \frac{1}{3} (x + 2)^{3} - 8 ln ∣ x + 2 ∣ + 12) + 5 (x + 2 - 2 ln ∣ x + 2 ∣)

Simplify

2 (12 x^{2} + 16 x + 16 ln ∣ x + 2 ∣ + \frac{1}{4} (x + 2)^{4} - \frac{8}{3} (x + 2)^{3} - 16) - 6 (- 3 x^{2} + \frac{1}{3} (x + 2)^{3} - 8 ln ∣ x + 2 ∣ + 12) + 5 (x + 2 - 2 ln ∣ x + 2 ∣) : 42 x^{2} + 37 x + 70 ln ∣ x + 2 ∣ + \frac{1}{2} (x + 2)^{4} - \frac{22}{3} (x + 2)^{3} - 94

= 42 x^{2} + 37 x + 70 ln ∣ x + 2 ∣ + \frac{1}{2} (x + 2)^{4} - \frac{22}{3} (x + 2)^{3} - 94

Add a constant to the solution

= 42 x^{2} + 37 x + 70 ln ∣ x + 2 ∣ + \frac{1}{2} (x + 2)^{4} - \frac{22}{3} (x + 2)^{3} - 94 + C

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What is the integral of (2x^4-6x^3+5x)/(x+2) ?
The integral of (2x^4-6x^3+5x)/(x+2) is 42x^2+37x+70ln|x+2|+1/2 (x+2)^4-22/3 (x+2)^3-94+C