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

The integral of (x^2)/((25+x^2)^2) is 1/10 arctan(x/5)-x/(2(25+x^2))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

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

Solution steps

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

Apply Integral Substitution

= \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

= \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}}

= \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

= \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}})

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

Substitute back

u = \frac{x}{5}

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

Simplify

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

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

Add a constant to the solution

= \frac{1}{10} arctan (\frac{x}{5}) - \frac{x}{2 ( 25 + x ^{2} )} + C

5 x y \frac{d y}{d x} = 1

y^{^{''}} - 3 y^{^{'}} - 12 y = 0

\frac{4 e ^{(- 2 s)}}{( s ^{2} - 4 s + 3 )}

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

\int \frac{arctan ( 9 x )}{1 + 81 x ^{2}} d x

What is the integral of (x^2)/((25+x^2)^2) ?
The integral of (x^2)/((25+x^2)^2) is 1/10 arctan(x/5)-x/(2(25+x^2))+C