What is the integral of (x^2)/((x+6)^3) ?

The integral of (x^2)/((x+6)^3) is -(x^2)/(2(x+6)^2)+ln|x+6|+6/(x+6)+C

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integral of (x^2)/((x+6)^3)

\int \frac{x ^{2}}{( x + 6 ) ^{3}} d x

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

Solution steps

\int \frac{x ^{2}}{( x + 6 ) ^{3}} d x

Apply Integration By Parts

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

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

\int - \frac{x}{( x + 6 ) ^{2}} d x = - ln ∣ x + 6 ∣ - \frac{6}{x + 6}

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

Simplify

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

Add a constant to the solution

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

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What is the integral of (x^2)/((x+6)^3) ?
The integral of (x^2)/((x+6)^3) is -(x^2)/(2(x+6)^2)+ln|x+6|+6/(x+6)+C