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

The integral of (x+1)/(x^2(x+2)^3) is -1/16 ln|x|-1/(8x)+1/16 ln|x+2|+1/(8(x+2)^2)+C

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

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

- \frac{1}{16} ln ∣ x ∣ - \frac{1}{8 x} + \frac{1}{16} ln ∣ x + 2 ∣ + \frac{1}{8 ( x + 2 ) ^{2}} + C

Solution steps

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

Apply u-substitution

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

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

Take the partial fraction of

\frac{u}{( u - 1 ) ^{2} ( u + 1 ) ^{3}} : - \frac{1}{16 ( u - 1 )} + \frac{1}{8 ( u - 1 ) ^{2}} + \frac{1}{16 ( u + 1 )} - \frac{1}{4 ( u + 1 ) ^{3}}

= \int - \frac{1}{16 ( u - 1 )} + \frac{1}{8 ( u - 1 ) ^{2}} + \frac{1}{16 ( u + 1 )} - \frac{1}{4 ( u + 1 ) ^{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{1}{16 ( u - 1 )} d u + \int \frac{1}{8 ( u - 1 ) ^{2}} d u + \int \frac{1}{16 ( u + 1 )} d u - \int \frac{1}{4 ( u + 1 ) ^{3}} d u

\int \frac{1}{16 ( u - 1 )} d u = \frac{1}{16} ln ∣ u - 1 ∣

\int \frac{1}{8 ( u - 1 ) ^{2}} d u = - \frac{1}{8 ( u - 1 )}

\int \frac{1}{16 ( u + 1 )} d u = \frac{1}{16} ln ∣ u + 1 ∣

\int \frac{1}{4 ( u + 1 ) ^{3}} d u = - \frac{1}{8 ( u + 1 ) ^{2}}

= - \frac{1}{16} ln ∣ u - 1 ∣ - \frac{1}{8 ( u - 1 )} + \frac{1}{16} ln ∣ u + 1 ∣ - (- \frac{1}{8 ( u + 1 ) ^{2}})

Substitute back

u = x + 1

= - \frac{1}{16} ln ∣ x + 1 - 1 ∣ - \frac{1}{8 ( x + 1 - 1 )} + \frac{1}{16} ln ∣ x + 1 + 1 ∣ - (- \frac{1}{8 ( x + 1 + 1 ) ^{2}})

Simplify

- \frac{1}{16} ln ∣ x + 1 - 1 ∣ - \frac{1}{8 ( x + 1 - 1 )} + \frac{1}{16} ln ∣ x + 1 + 1 ∣ - (- \frac{1}{8 ( x + 1 + 1 ) ^{2}}) : - \frac{1}{16} ln ∣ x ∣ - \frac{1}{8 x} + \frac{1}{16} ln ∣ x + 2 ∣ + \frac{1}{8 ( x + 2 ) ^{2}}

= - \frac{1}{16} ln ∣ x ∣ - \frac{1}{8 x} + \frac{1}{16} ln ∣ x + 2 ∣ + \frac{1}{8 ( x + 2 ) ^{2}}

Add a constant to the solution

= - \frac{1}{16} ln ∣ x ∣ - \frac{1}{8 x} + \frac{1}{16} ln ∣ x + 2 ∣ + \frac{1}{8 ( x + 2 ) ^{2}} + C

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What is the integral of (x+1)/(x^2(x+2)^3) ?
The integral of (x+1)/(x^2(x+2)^3) is -1/16 ln|x|-1/(8x)+1/16 ln|x+2|+1/(8(x+2)^2)+C