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

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

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

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

\frac{1}{x ^{2} + 1} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= - 2 \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

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

Apply the Power Rule

: - \frac{1}{u}

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

Substitute back

u = x^{2} + 1

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

Simplify

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

= \frac{1}{x ^{2} + 1}

Add a constant to the solution

= \frac{1}{x ^{2} + 1} + C

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