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

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

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integral of r/((1+r^2)^2)

integral

\frac{r}{( 1 + r ^{2} ) ^{2}}

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

Solution steps

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

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 = 1 + r^{2}

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

Simplify

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

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

Add a constant to the solution

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

sin^{8} (x)

\frac{1}{1 + x}

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

x + 3

\frac{4 x ^{2}}{x ^{2} + 2}

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