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

The integral of (3x^2-2)/((x+2)^3) is 3ln|x+2|+(12)/(x+2)-5/((x+2)^2)+C

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

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

3 ln ∣ x + 2 ∣ + \frac{12}{x + 2} - \frac{5}{( x + 2 ) ^{2}} + C

Solution steps

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

Apply u-substitution

: \int \frac{3 u ^{2} - 12 u + 10}{u ^{3}} d u

= \int \frac{3 u ^{2} - 12 u + 10}{u ^{3}} d u

Expand

\frac{3 u ^{2} - 12 u + 10}{u ^{3}} : \frac{3}{u} - \frac{12}{u ^{2}} + \frac{10}{u ^{3}}

= \int \frac{3}{u} - \frac{12}{u ^{2}} + \frac{10}{u ^{3}} d u

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{3}{u} d u - \int \frac{12}{u ^{2}} d u + \int \frac{10}{u ^{3}} d u

\int \frac{3}{u} d u = 3 ln ∣ u ∣

\int \frac{12}{u ^{2}} d u = - \frac{12}{u}

\int \frac{10}{u ^{3}} d u = - \frac{5}{u ^{2}}

= 3 ln ∣ u ∣ - (- \frac{12}{u}) - \frac{5}{u ^{2}}

Substitute back

u = x + 2

= 3 ln ∣ x + 2 ∣ - (- \frac{12}{x + 2}) - \frac{5}{( x + 2 ) ^{2}}

Simplify

= 3 ln ∣ x + 2 ∣ + \frac{12}{x + 2} - \frac{5}{( x + 2 ) ^{2}}

Add a constant to the solution

= 3 ln ∣ x + 2 ∣ + \frac{12}{x + 2} - \frac{5}{( x + 2 ) ^{2}} + C

\int_{1}^{x} f (t) d t

x y (d x - d y) = x^{2} d y + y^{2} d x

y^{^{''}} + 8 y^{^{'}} + 17 y = 0

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

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

What is the integral of (3x^2-2)/((x+2)^3) ?
The integral of (3x^2-2)/((x+2)^3) is 3ln|x+2|+(12)/(x+2)-5/((x+2)^2)+C