What is the integral of (x^2+2x+4)/((x+1)^3) ?

The integral of (x^2+2x+4)/((x+1)^3) is

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integral of (x^2+2x+4)/((x+1)^3)

\int \frac{x ^{2} + 2 x + 4}{( x + 1 ) ^{3}} d x

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

Solution steps

\int \frac{x ^{2} + 2 x + 4}{( x + 1 ) ^{3}} d x

Expand

\frac{x ^{2} + 2 x + 4}{( x + 1 ) ^{3}} : \frac{x ^{2}}{( x + 1 ) ^{3}} + \frac{2 x}{( x + 1 ) ^{3}} + \frac{4}{( x + 1 ) ^{3}}

= \int \frac{x ^{2}}{( x + 1 ) ^{3}} + \frac{2 x}{( x + 1 ) ^{3}} + \frac{4}{( x + 1 ) ^{3}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x ^{2}}{( x + 1 ) ^{3}} d x + \int \frac{2 x}{( x + 1 ) ^{3}} d x + \int \frac{4}{( x + 1 ) ^{3}} d x

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

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

\int \frac{4}{( x + 1 ) ^{3}} d x = - \frac{2}{( x + 1 ) ^{2}}

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

Simplify

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

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

Add a constant to the solution

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

\int 5 tan^{2} (x) d x

x + x

x \to a lim (f (x) (g (x)))

\frac{\partial}{\partial θ} (r (θ))

\frac{d}{d x} ((x^{2} - 2 x) e^{4 x})

What is the integral of (x^2+2x+4)/((x+1)^3) ?
The integral of (x^2+2x+4)/((x+1)^3) is