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

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

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

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

- \frac{x ^{2}}{2 ( x + 3 ) ^{2}} + ln ∣ x + 3 ∣ + \frac{3}{x + 3} + C

Solution steps

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

Apply Integration By Parts

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

\int - \frac{x}{( x + 3 ) ^{2}} d x = - ln ∣ x + 3 ∣ - \frac{3}{x + 3}

= - \frac{x ^{2}}{2 ( x + 3 ) ^{2}} - (- ln ∣ x + 3 ∣ - \frac{3}{x + 3})

Simplify

= - \frac{x ^{2}}{2 ( x + 3 ) ^{2}} + ln ∣ x + 3 ∣ + \frac{3}{x + 3}

Add a constant to the solution

= - \frac{x ^{2}}{2 ( x + 3 ) ^{2}} + ln ∣ x + 3 ∣ + \frac{3}{x + 3} + C

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