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

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

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

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

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

Solution steps

\int \frac{- 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

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

Apply u-substitution

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

= - \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

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

Apply the Power Rule

: - \frac{1}{u}

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

Substitute back

u = x^{2} + 1

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

Simplify

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

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

Add a constant to the solution

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

x \to - 3 lim (\frac{3 - ∣ x ∣}{x + 3})

y^{^{''}} + 17 y^{^{'}} + 70 y = 0

(- 5, - 2), (3, - 9)

\int_{0}^{2} (10 x - 5 x^{2}) d x

\int \frac{1}{10 - 0.8 x} d x

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