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

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

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

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

\frac{3}{2 ( x ^{2} + 2 )} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{2 u ^{2}} d u

= - 3 \cdot \int \frac{1}{2 u ^{2}} d u

Take the constant out:

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

= - 3 \cdot \frac{1}{2} \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

= - 3 \cdot \frac{1}{2} (- \frac{1}{u})

Substitute back

u = x^{2} + 2

= - 3 \cdot \frac{1}{2} (- \frac{1}{x ^{2} + 2})

Simplify

- 3 \cdot \frac{1}{2} (- \frac{1}{x ^{2} + 2}) : \frac{3}{2 ( x ^{2} + 2 )}

= \frac{3}{2 ( x ^{2} + 2 )}

Add a constant to the solution

= \frac{3}{2 ( x ^{2} + 2 )} + C

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