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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int - \frac{1}{v ^{2}} d v

= \frac{1}{2} \cdot \int - \frac{1}{v ^{2}} d v

Take the constant out:

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

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

Apply the Power Rule

: - \frac{1}{v}

= \frac{1}{2} (- (- \frac{1}{v}))

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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