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

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

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

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

- \frac{x ^{2} - 3}{2 ( x + 1 ) ^{2}} + ln ∣ x + 1 ∣ + \frac{1}{x + 1} + C

Solution steps

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

Apply Integration By Parts

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

= - \frac{x ^{2} - 3}{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} - 3}{2 ( x + 1 ) ^{2}} - (- ln ∣ x + 1 ∣ - \frac{1}{x + 1})

Simplify

= - \frac{x ^{2} - 3}{2 ( x + 1 ) ^{2}} + ln ∣ x + 1 ∣ + \frac{1}{x + 1}

Add a constant to the solution

= - \frac{x ^{2} - 3}{2 ( x + 1 ) ^{2}} + ln ∣ x + 1 ∣ + \frac{1}{x + 1} + C

y^{^{''}} - 6 y^{^{'}} + 25 y = 3 t - 2 cos (5 t)

\int (6 e^{7 x}) d x

\int \frac{sin ( a ) x}{c + cos ( a ) x} d x

\int \frac{1}{7} x^{- \frac{6}{7}} d x

y^{2} + x y + x^{2} = 1, (0, 1)

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