What is the integral of (x^2)/(x-7) ?

The integral of (x^2)/(x-7) is ((x-7)^2)/2+14(x-7)+49ln|x-7|+C

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integral of (x^2)/(x-7)

\int \frac{x ^{2}}{x - 7} d x

\frac{( x - 7 ) ^{2}}{2} + 14 (x - 7) + 49 ln ∣ x - 7 ∣ + C

Solution steps

\int \frac{x ^{2}}{x - 7} d x

Apply u-substitution

: \int \frac{( u + 7 ) ^{2}}{u} d u

= \int \frac{( u + 7 ) ^{2}}{u} d u

Expand

\frac{( u + 7 ) ^{2}}{u} : u + 14 + \frac{49}{u}

= \int u + 14 + \frac{49}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u d u + \int 14 d u + \int \frac{49}{u} d u

\int u d u = \frac{u ^{2}}{2}

\int 14 d u = 14 u

\int \frac{49}{u} d u = 49 ln ∣ u ∣

= \frac{u ^{2}}{2} + 14 u + 49 ln ∣ u ∣

Substitute back

u = x - 7

= \frac{( x - 7 ) ^{2}}{2} + 14 (x - 7) + 49 ln ∣ x - 7 ∣

Add a constant to the solution

= \frac{( x - 7 ) ^{2}}{2} + 14 (x - 7) + 49 ln ∣ x - 7 ∣ + C

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What is the integral of (x^2)/(x-7) ?
The integral of (x^2)/(x-7) is ((x-7)^2)/2+14(x-7)+49ln|x-7|+C