What is the integral of 1/((x^3-8)^2) ?

The integral of 1/((x^3-8)^2) is

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integral of 1/((x^3-8)^2)

\int \frac{1}{( x ^{3} - 8 ) ^{2}} d x

- \frac{1}{144} ln ∣ x - 2 ∣ + \frac{1}{288} ln \frac{1}{3} (x + 1)^{2} + 1 + \frac{1}{48 3} arctan (\frac{1}{3} (x + 1)) + \frac{1}{288 3} sin (2 arctan (\frac{1}{3} (x + 1))) - \frac{1}{144 ( x - 2 )} - \frac{1}{48 ( x ^{2} + 2 x + 4 )} + C

Solution steps

\int \frac{1}{( x ^{3} - 8 ) ^{2}} d x

Take the partial fraction of

\frac{1}{( x ^{3} - 8 ) ^{2}} : - \frac{1}{144 ( x - 2 )} + \frac{1}{144 ( x - 2 ) ^{2}} + \frac{x + 3}{144 ( x ^{2} + 2 x + 4 )} + \frac{x + 2}{24 ( x ^{2} + 2 x + 4 ) ^{2}}

= \int - \frac{1}{144 ( x - 2 )} + \frac{1}{144 ( x - 2 ) ^{2}} + \frac{x + 3}{144 ( x ^{2} + 2 x + 4 )} + \frac{x + 2}{24 ( x ^{2} + 2 x + 4 ) ^{2}} 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{1}{144 ( x - 2 )} d x + \int \frac{1}{144 ( x - 2 ) ^{2}} d x + \int \frac{x + 3}{144 ( x ^{2} + 2 x + 4 )} d x + \int \frac{x + 2}{24 ( x ^{2} + 2 x + 4 ) ^{2}} d x

\int \frac{1}{144 ( x - 2 )} d x = \frac{1}{144} ln ∣ x - 2 ∣

\int \frac{1}{144 ( x - 2 ) ^{2}} d x = - \frac{1}{144 ( x - 2 )}

\int \frac{x + 3}{144 ( x ^{2} + 2 x + 4 )} d x = \frac{1}{288} ln \frac{( x + 1 ) ^{2}}{3} + 1 + \frac{1}{72 3} arctan (\frac{1}{3} (x + 1))

\int \frac{x + 2}{24 ( x ^{2} + 2 x + 4 ) ^{2}} d x = \frac{1}{24} (- \frac{1}{2 ( x ^{2} + 2 x + 4 )} + \frac{1}{6 3} (arctan (\frac{1}{3} (x + 1)) + \frac{1}{2} sin (2 arctan (\frac{1}{3} (x + 1)))))

= - \frac{1}{144} ln ∣ x - 2 ∣ - \frac{1}{144 ( x - 2 )} + \frac{1}{288} ln \frac{( x + 1 ) ^{2}}{3} + 1 + \frac{1}{72 3} arctan (\frac{1}{3} (x + 1)) + \frac{1}{24} (- \frac{1}{2 ( x ^{2} + 2 x + 4 )} + \frac{1}{6 3} (arctan (\frac{1}{3} (x + 1)) + \frac{1}{2} sin (2 arctan (\frac{1}{3} (x + 1)))))

Simplify

- \frac{1}{144} ln ∣ x - 2 ∣ - \frac{1}{144 ( x - 2 )} + \frac{1}{288} ln \frac{( x + 1 ) ^{2}}{3} + 1 + \frac{1}{72 3} arctan (\frac{1}{3} (x + 1)) + \frac{1}{24} (- \frac{1}{2 ( x ^{2} + 2 x + 4 )} + \frac{1}{6 3} (arctan (\frac{1}{3} (x + 1)) + \frac{1}{2} sin (2 arctan (\frac{1}{3} (x + 1))))) : - \frac{1}{144} ln ∣ x - 2 ∣ + \frac{1}{288} ln \frac{1}{3} (x + 1)^{2} + 1 + \frac{1}{48 3} arctan (\frac{1}{3} (x + 1)) + \frac{1}{288 3} sin (2 arctan (\frac{1}{3} (x + 1))) - \frac{1}{144 ( x - 2 )} - \frac{1}{48 ( x ^{2} + 2 x + 4 )}

= - \frac{1}{144} ln ∣ x - 2 ∣ + \frac{1}{288} ln \frac{1}{3} (x + 1)^{2} + 1 + \frac{1}{48 3} arctan (\frac{1}{3} (x + 1)) + \frac{1}{288 3} sin (2 arctan (\frac{1}{3} (x + 1))) - \frac{1}{144 ( x - 2 )} - \frac{1}{48 ( x ^{2} + 2 x + 4 )}

Add a constant to the solution

= - \frac{1}{144} ln ∣ x - 2 ∣ + \frac{1}{288} ln \frac{1}{3} (x + 1)^{2} + 1 + \frac{1}{48 3} arctan (\frac{1}{3} (x + 1)) + \frac{1}{288 3} sin (2 arctan (\frac{1}{3} (x + 1))) - \frac{1}{144 ( x - 2 )} - \frac{1}{48 ( x ^{2} + 2 x + 4 )} + C

\int \frac{y ^{2} - 25}{y} d y

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

\int x cos^{2} (2 x) d x

\frac{( - 1 )}{s ^{2} + 5 s + 6}

f (x) = \frac{3 x - 1}{x}