What is the integral of 4/((x^2-16)^2) ?

The integral of 4/((x^2-16)^2) is 4(1/256 ln|x+4|-1/(64(x+4))-1/256 ln|x-4|-1/(64(x-4)))+C

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integral of 4/((x^2-16)^2)

\int \frac{4}{( x ^{2} - 16 ) ^{2}} d x

4 (\frac{1}{256} ln ∣ x + 4 ∣ - \frac{1}{64 ( x + 4 )} - \frac{1}{256} ln ∣ x - 4 ∣ - \frac{1}{64 ( x - 4 )}) + C

Solution steps

\int \frac{4}{( x ^{2} - 16 ) ^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int \frac{1}{( x ^{2} - 16 ) ^{2}} d x

Take the partial fraction of

\frac{1}{( x ^{2} - 16 ) ^{2}} : \frac{1}{256 ( x + 4 )} + \frac{1}{64 ( x + 4 ) ^{2}} - \frac{1}{256 ( x - 4 )} + \frac{1}{64 ( x - 4 ) ^{2}}

= 4 \cdot \int \frac{1}{256 ( x + 4 )} + \frac{1}{64 ( x + 4 ) ^{2}} - \frac{1}{256 ( x - 4 )} + \frac{1}{64 ( 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

= 4 (\int \frac{1}{256 ( x + 4 )} d x + \int \frac{1}{64 ( x + 4 ) ^{2}} d x - \int \frac{1}{256 ( x - 4 )} d x + \int \frac{1}{64 ( x - 4 ) ^{2}} d x)

\int \frac{1}{256 ( x + 4 )} d x = \frac{1}{256} ln ∣ x + 4 ∣

\int \frac{1}{64 ( x + 4 ) ^{2}} d x = - \frac{1}{64 ( x + 4 )}

\int \frac{1}{256 ( x - 4 )} d x = \frac{1}{256} ln ∣ x - 4 ∣

\int \frac{1}{64 ( x - 4 ) ^{2}} d x = - \frac{1}{64 ( x - 4 )}

= 4 (\frac{1}{256} ln ∣ x + 4 ∣ - \frac{1}{64 ( x + 4 )} - \frac{1}{256} ln ∣ x - 4 ∣ - \frac{1}{64 ( x - 4 )})

Add a constant to the solution

= 4 (\frac{1}{256} ln ∣ x + 4 ∣ - \frac{1}{64 ( x + 4 )} - \frac{1}{256} ln ∣ x - 4 ∣ - \frac{1}{64 ( x - 4 )}) + C

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What is the integral of 4/((x^2-16)^2) ?
The integral of 4/((x^2-16)^2) is 4(1/256 ln|x+4|-1/(64(x+4))-1/256 ln|x-4|-1/(64(x-4)))+C