What is the integral of (x^2)/(6+x) ?

The integral of (x^2)/(6+x) is ((6+x)^2)/2-12(6+x)+36ln|6+x|+C

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integral of (x^2)/(6+x)

\int \frac{x ^{2}}{6 + x} d x

\frac{( 6 + x ) ^{2}}{2} - 12 (6 + x) + 36 ln ∣ 6 + x ∣ + C

Solution steps

\int \frac{x ^{2}}{6 + x} d x

Apply u-substitution

: \int \frac{( u - 6 ) ^{2}}{u} d u

= \int \frac{( u - 6 ) ^{2}}{u} d u

Expand

\frac{( u - 6 ) ^{2}}{u} : u - 12 + \frac{36}{u}

= \int u - 12 + \frac{36}{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 12 d u + \int \frac{36}{u} d u

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

\int 12 d u = 12 u

\int \frac{36}{u} d u = 36 ln ∣ u ∣

= \frac{u ^{2}}{2} - 12 u + 36 ln ∣ u ∣

Substitute back

u = 6 + x

= \frac{( 6 + x ) ^{2}}{2} - 12 (6 + x) + 36 ln ∣ 6 + x ∣

Add a constant to the solution

= \frac{( 6 + x ) ^{2}}{2} - 12 (6 + x) + 36 ln ∣ 6 + x ∣ + C

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What is the integral of (x^2)/(6+x) ?
The integral of (x^2)/(6+x) is ((6+x)^2)/2-12(6+x)+36ln|6+x|+C