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

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

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

integral

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

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

Solution steps

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

Take the partial fraction of

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

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

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

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

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

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

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

Add a constant to the solution

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

\frac{x ^{2}}{( ( x - 2 ) ( x ^{2} + 1 ) )}

1 + y^{2}

(x + 1) arctan (x)

\frac{1}{( 2 x + 1 )}

\frac{1}{( x \cdot 1 + x ^{2} )}

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