What is the integral of (x^3-4x+6)/(x+7) ?

The integral of (x^3-4x+6)/(x+7) is x^3-4x+21/2 (x+7)^2-309ln|x+7|-2/3 (x+7)^3+6+C

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integral of (x^3-4x+6)/(x+7)

\int \frac{x ^{3} - 4 x + 6}{x + 7} d x

x^{3} - 4 x + \frac{21}{2} (x + 7)^{2} - 309 ln ∣ x + 7 ∣ - \frac{2}{3} (x + 7)^{3} + 6 + C

Solution steps

\int \frac{x ^{3} - 4 x + 6}{x + 7} d x

Apply u-substitution

: \int \frac{- 4 u + ( u - 7 ) ^{3} + 34}{u} d u

= \int \frac{- 4 u + ( u - 7 ) ^{3} + 34}{u} d u

Apply Integration By Parts

: - 4 u + (u - 7)^{3} + 34 - \int \frac{3 u ^{3} - 42 u ^{2} + 147 u - ( u - 7 ) ^{3} - 34}{u} d u

= - 4 u + (u - 7)^{3} + 34 - \int \frac{3 u ^{3} - 42 u ^{2} + 147 u - ( u - 7 ) ^{3} - 34}{u} d u

\int \frac{3 u ^{3} - 42 u ^{2} + 147 u - ( u - 7 ) ^{3} - 34}{u} d u = \frac{2 u ^{3}}{3} - \frac{21 u ^{2}}{2} + 309 ln ∣ u ∣

= - 4 u + (u - 7)^{3} + 34 - (\frac{2 u ^{3}}{3} - \frac{21 u ^{2}}{2} + 309 ln ∣ u ∣)

Substitute back

u = x + 7

= - 4 (x + 7) + (x + 7 - 7)^{3} + 34 - (\frac{2 ( x + 7 ) ^{3}}{3} - \frac{21 ( x + 7 ) ^{2}}{2} + 309 ln ∣ x + 7 ∣)

Simplify

- 4 (x + 7) + (x + 7 - 7)^{3} + 34 - (\frac{2 ( x + 7 ) ^{3}}{3} - \frac{21 ( x + 7 ) ^{2}}{2} + 309 ln ∣ x + 7 ∣) : x^{3} - 4 x + \frac{21}{2} (x + 7)^{2} - 309 ln ∣ x + 7 ∣ - \frac{2}{3} (x + 7)^{3} + 6

= x^{3} - 4 x + \frac{21}{2} (x + 7)^{2} - 309 ln ∣ x + 7 ∣ - \frac{2}{3} (x + 7)^{3} + 6

Add a constant to the solution

= x^{3} - 4 x + \frac{21}{2} (x + 7)^{2} - 309 ln ∣ x + 7 ∣ - \frac{2}{3} (x + 7)^{3} + 6 + C

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What is the integral of (x^3-4x+6)/(x+7) ?
The integral of (x^3-4x+6)/(x+7) is x^3-4x+21/2 (x+7)^2-309ln|x+7|-2/3 (x+7)^3+6+C