What is the integral of (x^2)/(x+5) ?

The integral of (x^2)/(x+5) is ((x+5)^2)/2-10(x+5)+25ln|x+5|+C

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integral of (x^2)/(x+5)

\int \frac{x ^{2}}{x + 5} d x

\frac{( x + 5 ) ^{2}}{2} - 10 (x + 5) + 25 ln ∣ x + 5 ∣ + C

Solution steps

\int \frac{x ^{2}}{x + 5} d x

Apply u-substitution

= \int \frac{( u - 5 ) ^{2}}{u} d u

Expand

\frac{( u - 5 ) ^{2}}{u} : u - 10 + \frac{25}{u}

= \int u - 10 + \frac{25}{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 10 d u + \int \frac{25}{u} d u

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

\int 10 d u = 10 u

\int \frac{25}{u} d u = 25 ln ∣ u ∣

= \frac{u ^{2}}{2} - 10 u + 25 ln ∣ u ∣

Substitute back

u = x + 5

= \frac{( x + 5 ) ^{2}}{2} - 10 (x + 5) + 25 ln ∣ x + 5 ∣

Add a constant to the solution

= \frac{( x + 5 ) ^{2}}{2} - 10 (x + 5) + 25 ln ∣ x + 5 ∣ + C

What is the integral of (x^2)/(x+5) ?
The integral of (x^2)/(x+5) is ((x+5)^2)/2-10(x+5)+25ln|x+5|+C