What is the integral of (x^2)/((4x^2+9)^2) ?

The integral of (x^2)/((4x^2+9)^2) is 1/24 (1/2 arctan(2/3 x)-(3x)/(9+4x^2))+C

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integral of (x^2)/((4x^2+9)^2)

\int \frac{x ^{2}}{( 4 x ^{2} + 9 ) ^{2}} d x

\frac{1}{24} (\frac{1}{2} arctan (\frac{2}{3} x) - \frac{3 x}{9 + 4 x ^{2}}) + C

Solution steps

\int \frac{x ^{2}}{( 4 x ^{2} + 9 ) ^{2}} d x

Apply Integral Substitution

= \int \frac{u ^{2}}{24 ( u ^{2} + 1 ) ^{2}} d u

Take the constant out:

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

= \frac{1}{24} \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}}

= \frac{1}{24} \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

= \frac{1}{24} (\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}})

= \frac{1}{24} (arctan (u) - \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}}))

Substitute back

u = \frac{2}{3} x

= \frac{1}{24} (arctan (\frac{2}{3} x) - \frac{1}{2} (arctan (\frac{2}{3} x) + \frac{\frac{2}{3} x}{1 + ( \frac{2}{3} x ) ^{2}}))

Simplify

\frac{1}{24} (arctan (\frac{2}{3} x) - \frac{1}{2} (arctan (\frac{2}{3} x) + \frac{\frac{2}{3} x}{1 + ( \frac{2}{3} x ) ^{2}})) : \frac{1}{24} (\frac{1}{2} arctan (\frac{2}{3} x) - \frac{3 x}{9 + 4 x ^{2}})

= \frac{1}{24} (\frac{1}{2} arctan (\frac{2}{3} x) - \frac{3 x}{9 + 4 x ^{2}})

Add a constant to the solution

= \frac{1}{24} (\frac{1}{2} arctan (\frac{2}{3} x) - \frac{3 x}{9 + 4 x ^{2}}) + C

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What is the integral of (x^2)/((4x^2+9)^2) ?
The integral of (x^2)/((4x^2+9)^2) is 1/24 (1/2 arctan(2/3 x)-(3x)/(9+4x^2))+C