What is the integral of (5r)/((4+r^2)^2) ?

The integral of (5r)/((4+r^2)^2) is -5/(2(4+r^2))+C

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integral of (5r)/((4+r^2)^2)

\int \frac{5 r}{( 4 + r ^{2} ) ^{2}} d r

- \frac{5}{2 ( 4 + r ^{2} )} + C

Solution steps

\int \frac{5 r}{( 4 + r ^{2} ) ^{2}} d r

Take the constant out:

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

= 5 \cdot \int \frac{r}{( 4 + r ^{2} ) ^{2}} d r

Apply u-substitution

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

= 5 \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

= 5 \cdot \frac{1}{2} \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

= 5 \cdot \frac{1}{2} (- \frac{1}{u})

Substitute back

u = 4 + r^{2}

= 5 \cdot \frac{1}{2} (- \frac{1}{4 + r ^{2}})

Simplify

5 \cdot \frac{1}{2} (- \frac{1}{4 + r ^{2}}) : - \frac{5}{2 ( 4 + r ^{2} )}

= - \frac{5}{2 ( 4 + r ^{2} )}

Add a constant to the solution

= - \frac{5}{2 ( 4 + r ^{2} )} + C

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What is the integral of (5r)/((4+r^2)^2) ?
The integral of (5r)/((4+r^2)^2) is -5/(2(4+r^2))+C