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)

integral

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

- \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

= - \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

arctan (\frac{1}{2 x})

\frac{( x ^{2} + 4 )}{( x ^{2} - 3 x + 2 )}

\frac{e}{e ^{(x)^{2}}}

\frac{( x - 2 )}{( 2 x + 2 )}

\frac{x ^{3}}{e ^{x}}

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