What is the integral of (x^2)/((x+5)^3) ?

The integral of (x^2)/((x+5)^3) is -(x^2)/(2(x+5)^2)+ln|x+5|+5/(x+5)+C

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integral of (x^2)/((x+5)^3)

\int \frac{x ^{2}}{( x + 5 ) ^{3}} d x

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

Solution steps

\int \frac{x ^{2}}{( x + 5 ) ^{3}} d x

Apply Integration By Parts

: - \frac{x ^{2}}{2 ( x + 5 ) ^{2}} - \int - \frac{x}{( x + 5 ) ^{2}} d x

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

\int - \frac{x}{( x + 5 ) ^{2}} d x = - ln ∣ x + 5 ∣ - \frac{5}{x + 5}

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

Simplify

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

Add a constant to the solution

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

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What is the integral of (x^2)/((x+5)^3) ?
The integral of (x^2)/((x+5)^3) is -(x^2)/(2(x+5)^2)+ln|x+5|+5/(x+5)+C