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

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

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

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

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

Solution steps

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

Apply u-substitution

= \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 = 7 + x

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

Add a constant to the solution

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

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