What is the integral of (x^2)/(x+15) ?

The integral of (x^2)/(x+15) is ((x+15)^2)/2-30(x+15)+225ln|x+15|+C

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integral of (x^2)/(x+15)

\int \frac{x ^{2}}{x + 15} d x

\frac{( x + 15 ) ^{2}}{2} - 30 (x + 15) + 225 ln ∣ x + 15 ∣ + C

Solution steps

\int \frac{x ^{2}}{x + 15} d x

Apply u-substitution

= \int \frac{( u - 15 ) ^{2}}{u} d u

Expand

\frac{( u - 15 ) ^{2}}{u} : u - 30 + \frac{225}{u}

= \int u - 30 + \frac{225}{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 30 d u + \int \frac{225}{u} d u

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

\int 30 d u = 30 u

\int \frac{225}{u} d u = 225 ln ∣ u ∣

= \frac{u ^{2}}{2} - 30 u + 225 ln ∣ u ∣

Substitute back

u = x + 15

= \frac{( x + 15 ) ^{2}}{2} - 30 (x + 15) + 225 ln ∣ x + 15 ∣

Add a constant to the solution

= \frac{( x + 15 ) ^{2}}{2} - 30 (x + 15) + 225 ln ∣ x + 15 ∣ + C

What is the integral of (x^2)/(x+15) ?
The integral of (x^2)/(x+15) is ((x+15)^2)/2-30(x+15)+225ln|x+15|+C