What is the integral of (x^2-5)/((1+x)^3) ?

The integral of (x^2-5)/((1+x)^3) is -(x^2-5)/(2(1+x)^2)+ln|1+x|+1/(1+x)+C

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integral of (x^2-5)/((1+x)^3)

\int \frac{x ^{2} - 5}{( 1 + x ) ^{3}} d x

- \frac{x ^{2} - 5}{2 ( 1 + x ) ^{2}} + ln ∣ 1 + x ∣ + \frac{1}{1 + x} + C

Solution steps

\int \frac{x ^{2} - 5}{( 1 + x ) ^{3}} d x

Apply Integration By Parts

= - \frac{x ^{2} - 5}{2 ( 1 + x ) ^{2}} - \int - \frac{x}{( 1 + x ) ^{2}} d x

\int - \frac{x}{( 1 + x ) ^{2}} d x = - ln ∣ 1 + x ∣ - \frac{1}{1 + x}

= - \frac{x ^{2} - 5}{2 ( 1 + x ) ^{2}} - (- ln ∣ 1 + x ∣ - \frac{1}{1 + x})

Simplify

= - \frac{x ^{2} - 5}{2 ( 1 + x ) ^{2}} + ln ∣ 1 + x ∣ + \frac{1}{1 + x}

Add a constant to the solution

= - \frac{x ^{2} - 5}{2 ( 1 + x ) ^{2}} + ln ∣ 1 + x ∣ + \frac{1}{1 + x} + C

What is the integral of (x^2-5)/((1+x)^3) ?
The integral of (x^2-5)/((1+x)^3) is -(x^2-5)/(2(1+x)^2)+ln|1+x|+1/(1+x)+C