What is the integral of 1/((x^2-16)^3) ?

The integral of 1/((x^2-16)^3) is

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integral of 1/((x^2-16)^3)

\int \frac{1}{( x ^{2} - 16 ) ^{3}} d x

- \frac{3}{16384} ln ∣ x + 4 ∣ + \frac{3}{4096 ( x + 4 )} + \frac{1}{1024 ( x + 4 ) ^{2}} + \frac{3}{16384} ln ∣ x - 4 ∣ + \frac{3}{4096 ( x - 4 )} - \frac{1}{1024 ( x - 4 ) ^{2}} + C

Solution steps

\int \frac{1}{( x ^{2} - 16 ) ^{3}} d x

Take the partial fraction of

\frac{1}{( x ^{2} - 16 ) ^{3}} : - \frac{3}{16384 ( x + 4 )} - \frac{3}{4096 ( x + 4 ) ^{2}} - \frac{1}{512 ( x + 4 ) ^{3}} + \frac{3}{16384 ( x - 4 )} - \frac{3}{4096 ( x - 4 ) ^{2}} + \frac{1}{512 ( x - 4 ) ^{3}}

= \int - \frac{3}{16384 ( x + 4 )} - \frac{3}{4096 ( x + 4 ) ^{2}} - \frac{1}{512 ( x + 4 ) ^{3}} + \frac{3}{16384 ( x - 4 )} - \frac{3}{4096 ( x - 4 ) ^{2}} + \frac{1}{512 ( x - 4 ) ^{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{3}{16384 ( x + 4 )} d x - \int \frac{3}{4096 ( x + 4 ) ^{2}} d x - \int \frac{1}{512 ( x + 4 ) ^{3}} d x + \int \frac{3}{16384 ( x - 4 )} d x - \int \frac{3}{4096 ( x - 4 ) ^{2}} d x + \int \frac{1}{512 ( x - 4 ) ^{3}} d x

\int \frac{3}{16384 ( x + 4 )} d x = \frac{3}{16384} ln ∣ x + 4 ∣

\int \frac{3}{4096 ( x + 4 ) ^{2}} d x = - \frac{3}{4096 ( x + 4 )}

\int \frac{1}{512 ( x + 4 ) ^{3}} d x = - \frac{1}{1024 ( x + 4 ) ^{2}}

\int \frac{3}{16384 ( x - 4 )} d x = \frac{3}{16384} ln ∣ x - 4 ∣

\int \frac{3}{4096 ( x - 4 ) ^{2}} d x = - \frac{3}{4096 ( x - 4 )}

\int \frac{1}{512 ( x - 4 ) ^{3}} d x = - \frac{1}{1024 ( x - 4 ) ^{2}}

= - \frac{3}{16384} ln ∣ x + 4 ∣ - (- \frac{3}{4096 ( x + 4 )}) - (- \frac{1}{1024 ( x + 4 ) ^{2}}) + \frac{3}{16384} ln ∣ x - 4 ∣ - (- \frac{3}{4096 ( x - 4 )}) - \frac{1}{1024 ( x - 4 ) ^{2}}

Simplify

= - \frac{3}{16384} ln ∣ x + 4 ∣ + \frac{3}{4096 ( x + 4 )} + \frac{1}{1024 ( x + 4 ) ^{2}} + \frac{3}{16384} ln ∣ x - 4 ∣ + \frac{3}{4096 ( x - 4 )} - \frac{1}{1024 ( x - 4 ) ^{2}}

Add a constant to the solution

= - \frac{3}{16384} ln ∣ x + 4 ∣ + \frac{3}{4096 ( x + 4 )} + \frac{1}{1024 ( x + 4 ) ^{2}} + \frac{3}{16384} ln ∣ x - 4 ∣ + \frac{3}{4096 ( x - 4 )} - \frac{1}{1024 ( x - 4 ) ^{2}} + C