What is the integral of (-4x)/((2-2x^2)^3) ?

The integral of (-4x)/((2-2x^2)^3) is -1/(2(2-2x^2)^2)+C

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integral of (-4x)/((2-2x^2)^3)

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int - \frac{1}{4 u ^{3}} d u

= - 4 \cdot \int - \frac{1}{4 u ^{3}} d u

Take the constant out:

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

= - 4 (- \frac{1}{4} \cdot \int \frac{1}{u ^{3}} d u)

Apply the Power Rule

: - \frac{1}{2 u ^{2}}

= - 4 (- \frac{1}{4} (- \frac{1}{2 u ^{2}}))

Substitute back

u = 2 - 2 x^{2}

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of (-4x)/((2-2x^2)^3) ?
The integral of (-4x)/((2-2x^2)^3) is -1/(2(2-2x^2)^2)+C