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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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