What is the integral of (2x^2+16)/(x(x-2)^3) ?

The integral of (2x^2+16)/(x(x-2)^3) is 2/(x-2)-6/((x-2)^2)-2ln|x|+2ln|x-2|+C

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integral of (2x^2+16)/(x(x-2)^3)

\int \frac{2 x ^{2} + 16}{x ( x - 2 ) ^{3}} d x

\frac{2}{x - 2} - \frac{6}{( x - 2 ) ^{2}} - 2 ln ∣ x ∣ + 2 ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{2 x ^{2} + 16}{x ( x - 2 ) ^{3}} d x

Expand

\frac{2 x ^{2} + 16}{x ( x - 2 ) ^{3}} : \frac{2 x}{( x - 2 ) ^{3}} + \frac{16}{x ( x - 2 ) ^{3}}

= \int \frac{2 x}{( x - 2 ) ^{3}} + \frac{16}{x ( x - 2 ) ^{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{2 x}{( x - 2 ) ^{3}} d x + \int \frac{16}{x ( x - 2 ) ^{3}} d x

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

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

= 2 (- \frac{1}{x - 2} - \frac{1}{( x - 2 ) ^{2}}) + 16 (- \frac{1}{8} ln ∣ x ∣ + \frac{1}{8} ln ∣ x - 2 ∣ + \frac{1}{4 ( x - 2 )} - \frac{1}{4 ( x - 2 ) ^{2}})

Simplify

2 (- \frac{1}{x - 2} - \frac{1}{( x - 2 ) ^{2}}) + 16 (- \frac{1}{8} ln ∣ x ∣ + \frac{1}{8} ln ∣ x - 2 ∣ + \frac{1}{4 ( x - 2 )} - \frac{1}{4 ( x - 2 ) ^{2}}) : \frac{2}{x - 2} - \frac{6}{( x - 2 ) ^{2}} - 2 ln ∣ x ∣ + 2 ln ∣ x - 2 ∣

= \frac{2}{x - 2} - \frac{6}{( x - 2 ) ^{2}} - 2 ln ∣ x ∣ + 2 ln ∣ x - 2 ∣

Add a constant to the solution

= \frac{2}{x - 2} - \frac{6}{( x - 2 ) ^{2}} - 2 ln ∣ x ∣ + 2 ln ∣ x - 2 ∣ + C

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

x^{2} - 2

\int \frac{5 x}{( x ^{2} + 7 ) ^{2}} d x

\frac{d}{d x} (\frac{tan ( x )}{2})

\int_{1}^{2} (∣ x + 1 ∣ + 2) d x

What is the integral of (2x^2+16)/(x(x-2)^3) ?
The integral of (2x^2+16)/(x(x-2)^3) is 2/(x-2)-6/((x-2)^2)-2ln|x|+2ln|x-2|+C