What is the integral of (5x^2)/((9+x^2)^2) ?

The integral of (5x^2)/((9+x^2)^2) is (5(x^2arctan(x/3)-3x+9arctan(x/3)))/(6(9+x^2))+C

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integral of (5x^2)/((9+x^2)^2)

\int \frac{5 x ^{2}}{( 9 + x ^{2} ) ^{2}} d x

\frac{5 ( x ^{2} arctan ( \frac{x}{3} ) - 3 x + 9 arctan ( \frac{x}{3} ) )}{6 ( 9 + x ^{2} )} + C

Solution steps

\int \frac{5 x ^{2}}{( 9 + x ^{2} ) ^{2}} d x

Take the constant out:

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

= 5 \cdot \int \frac{x ^{2}}{( 9 + x ^{2} ) ^{2}} d x

Apply Integral Substitution

= 5 \cdot \int \frac{u ^{2}}{3 ( u ^{2} + 1 ) ^{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}{3} \cdot \int \frac{u ^{2}}{( u ^{2} + 1 ) ^{2}} d u

Take the partial fraction of

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

= 5 \cdot \frac{1}{3} \cdot \int \frac{1}{u ^{2} + 1} - \frac{1}{( u ^{2} + 1 ) ^{2}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 5 \cdot \frac{1}{3} (\int \frac{1}{u ^{2} + 1} d u - \int \frac{1}{( u ^{2} + 1 ) ^{2}} d u)

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

\int \frac{1}{( u ^{2} + 1 ) ^{2}} d u = \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}})

= 5 \cdot \frac{1}{3} (arctan (u) - \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}}))

Substitute back

u = \frac{x}{3}

= 5 \cdot \frac{1}{3} (arctan (\frac{x}{3}) - \frac{1}{2} (arctan (\frac{x}{3}) + \frac{\frac{x}{3}}{1 + ( \frac{x}{3} ) ^{2}}))

Simplify

5 \cdot \frac{1}{3} (arctan (\frac{x}{3}) - \frac{1}{2} (arctan (\frac{x}{3}) + \frac{\frac{x}{3}}{1 + ( \frac{x}{3} ) ^{2}})) : \frac{5 ( x ^{2} arctan ( \frac{x}{3} ) - 3 x + 9 arctan ( \frac{x}{3} ) )}{6 ( 9 + x ^{2} )}

= \frac{5 ( x ^{2} arctan ( \frac{x}{3} ) - 3 x + 9 arctan ( \frac{x}{3} ) )}{6 ( 9 + x ^{2} )}

Add a constant to the solution

= \frac{5 ( x ^{2} arctan ( \frac{x}{3} ) - 3 x + 9 arctan ( \frac{x}{3} ) )}{6 ( 9 + x ^{2} )} + C

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What is the integral of (5x^2)/((9+x^2)^2) ?
The integral of (5x^2)/((9+x^2)^2) is (5(x^2arctan(x/3)-3x+9arctan(x/3)))/(6(9+x^2))+C