What is the integral of (3x^2)/((25+x^2)^2) ?

The integral of (3x^2)/((25+x^2)^2) is (3(x^2arctan(x/5)-5x+25arctan(x/5)))/(10(25+x^2))+C

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integral of (3x^2)/((25+x^2)^2)

\int \frac{3 x ^{2}}{( 25 + x ^{2} ) ^{2}} d x

\frac{3 ( x ^{2} arctan ( \frac{x}{5} ) - 5 x + 25 arctan ( \frac{x}{5} ) )}{10 ( 25 + x ^{2} )} + C

Solution steps

\int \frac{3 x ^{2}}{( 25 + x ^{2} ) ^{2}} d x

Take the constant out:

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

= 3 \cdot \int \frac{x ^{2}}{( 25 + x ^{2} ) ^{2}} d x

Apply Integral Substitution

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

Take the constant out:

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

= 3 \cdot \frac{1}{5} \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}}

= 3 \cdot \frac{1}{5} \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

= 3 \cdot \frac{1}{5} (\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}})

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

Substitute back

u = \frac{x}{5}

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

Simplify

3 \cdot \frac{1}{5} (arctan (\frac{x}{5}) - \frac{1}{2} (arctan (\frac{x}{5}) + \frac{\frac{x}{5}}{1 + ( \frac{x}{5} ) ^{2}})) : \frac{3 ( x ^{2} arctan ( \frac{x}{5} ) - 5 x + 25 arctan ( \frac{x}{5} ) )}{10 ( 25 + x ^{2} )}

= \frac{3 ( x ^{2} arctan ( \frac{x}{5} ) - 5 x + 25 arctan ( \frac{x}{5} ) )}{10 ( 25 + x ^{2} )}

Add a constant to the solution

= \frac{3 ( x ^{2} arctan ( \frac{x}{5} ) - 5 x + 25 arctan ( \frac{x}{5} ) )}{10 ( 25 + x ^{2} )} + C

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

What is the integral of (3x^2)/((25+x^2)^2) ?
The integral of (3x^2)/((25+x^2)^2) is (3(x^2arctan(x/5)-5x+25arctan(x/5)))/(10(25+x^2))+C