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

The integral of ((x^2+5)/((x+2)^2)) is x+2-4ln|x+2|-9/(x+2)+C

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integral of ((x^2+5)/((x+2)^2))

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

x + 2 - 4 ln ∣ x + 2 ∣ - \frac{9}{x + 2} + C

Solution steps

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

Expand

\frac{x ^{2} + 5}{( x + 2 ) ^{2}} : \frac{x ^{2}}{( x + 2 ) ^{2}} + \frac{5}{( x + 2 ) ^{2}}

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

\int \frac{x ^{2}}{( x + 2 ) ^{2}} d x = x + 2 - 4 ln ∣ x + 2 ∣ - \frac{4}{x + 2}

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

= x + 2 - 4 ln ∣ x + 2 ∣ - \frac{4}{x + 2} - \frac{5}{x + 2}

Simplify

x + 2 - 4 ln ∣ x + 2 ∣ - \frac{4}{x + 2} - \frac{5}{x + 2} : x + 2 - 4 ln ∣ x + 2 ∣ - \frac{9}{x + 2}

= x + 2 - 4 ln ∣ x + 2 ∣ - \frac{9}{x + 2}

Add a constant to the solution

= x + 2 - 4 ln ∣ x + 2 ∣ - \frac{9}{x + 2} + C

\int \frac{x ^{3}}{x + 2} d x

\int y^{2} sin (y) d y

sin (2 x + 3)

x \to 0 lim (x^{3} cos (\frac{5}{x}))

\frac{\partial}{\partial y} (\frac{x}{y + 1})

What is the integral of ((x^2+5)/((x+2)^2)) ?
The integral of ((x^2+5)/((x+2)^2)) is x+2-4ln|x+2|-9/(x+2)+C