What is the integral of 9/((x^2-1)^2) ?

The integral of 9/((x^2-1)^2) is 9(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 9/((x^2-1)^2)

\int \frac{9}{( x ^{2} - 1 ) ^{2}} d x

9 (\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{9}{( x ^{2} - 1 ) ^{2}} d x

Take the constant out:

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

= 9 \cdot \int \frac{1}{( x ^{2} - 1 ) ^{2}} d x

(x^{2} - 1)^{2} = x^{4} - 2 x^{2} + 1

= 9 \cdot \int \frac{1}{x ^{4} - 2 x ^{2} + 1} d x

Take the partial fraction of

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

= 9 \cdot \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

= 9 (\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 )}

= 9 (\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

= 9 (\frac{1}{4} ln ∣ x + 1 ∣ - \frac{1}{4 ( x + 1 )} - \frac{1}{4} ln ∣ x - 1 ∣ - \frac{1}{4 ( x - 1 )}) + C

sec^{7} (x)

4 - x^{2}

\frac{d y}{d x} = 2 x

\frac{d}{d x} (3 x + 3)

\int x^{\frac{1}{8}} + y^{\frac{1}{8}}

What is the integral of 9/((x^2-1)^2) ?
The integral of 9/((x^2-1)^2) is 9(1/4 ln|x+1|-1/(4(x+1))-1/4 ln|x-1|-1/(4(x-1)))+C