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

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

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integral of (x^3)/(2-x)

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

- x^{3} - \frac{2}{3} (2 - x)^{3} + 3 (2 - x)^{2} - 8 ln ∣ 2 - x ∣ + C

Solution steps

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

Apply u-substitution

: \int - \frac{( - u + 2 ) ^{3}}{u} d u

= \int - \frac{( - u + 2 ) ^{3}}{u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{( - u + 2 ) ^{3}}{u} d u

Apply Integration By Parts

: (- u + 2)^{3} - \int \frac{- 3 u ( - u + 2 ) ^{2} - ( - u + 2 ) ^{3}}{u} d u

= - ((- u + 2)^{3} - \int \frac{- 3 u ( - u + 2 ) ^{2} - ( - u + 2 ) ^{3}}{u} d u)

\int \frac{- 3 u ( - u + 2 ) ^{2} - ( - u + 2 ) ^{3}}{u} d u = - \frac{2 u ^{3}}{3} + 3 u^{2} - 8 ln ∣ u ∣

= - ((- u + 2)^{3} - (- \frac{2 u ^{3}}{3} + 3 u^{2} - 8 ln ∣ u ∣))

Substitute back

u = 2 - x

= - ((- (2 - x) + 2)^{3} - (- \frac{2 ( 2 - x ) ^{3}}{3} + 3 (2 - x)^{2} - 8 ln ∣ 2 - x ∣))

Simplify

- ((- (2 - x) + 2)^{3} - (- \frac{2 ( 2 - x ) ^{3}}{3} + 3 (2 - x)^{2} - 8 ln ∣ 2 - x ∣)) : - x^{3} - \frac{2}{3} (2 - x)^{3} + 3 (2 - x)^{2} - 8 ln ∣ 2 - x ∣

= - x^{3} - \frac{2}{3} (2 - x)^{3} + 3 (2 - x)^{2} - 8 ln ∣ 2 - x ∣

Add a constant to the solution

= - x^{3} - \frac{2}{3} (2 - x)^{3} + 3 (2 - x)^{2} - 8 ln ∣ 2 - x ∣ + C

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