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

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 y^{^{'}} + y = 3 x y^{2}, y (1) = - 3

7 x

\frac{\partial}{\partial y} (- 4 x^{3} y)

\frac{d}{d x} (3 d x)

\frac{\partial}{\partial z} (ln (\frac{x y}{y ^{2} + z ^{2}}))

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