What is the integral of (5x^2)/((36+x^2)^2) ?

The integral of (5x^2)/((36+x^2)^2) is (5(x^2arctan(x/6)-6x+36arctan(x/6)))/(12(36+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 (5x^2)/((36+x^2)^2)

\int \frac{5 x ^{2}}{( 36 + x ^{2} ) ^{2}} d x

\frac{5 ( x ^{2} arctan ( \frac{x}{6} ) - 6 x + 36 arctan ( \frac{x}{6} ) )}{12 ( 36 + x ^{2} )} + C

Solution steps

\int \frac{5 x ^{2}}{( 36 + 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}}{( 36 + x ^{2} ) ^{2}} d x

Apply Integral Substitution

= 5 \cdot \int \frac{u ^{2}}{6 ( 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}{6} \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}{6} \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}{6} (\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}{6} (arctan (u) - \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}}))

Substitute back

u = \frac{x}{6}

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

Simplify

5 \cdot \frac{1}{6} (arctan (\frac{x}{6}) - \frac{1}{2} (arctan (\frac{x}{6}) + \frac{\frac{x}{6}}{1 + ( \frac{x}{6} ) ^{2}})) : \frac{5 ( x ^{2} arctan ( \frac{x}{6} ) - 6 x + 36 arctan ( \frac{x}{6} ) )}{12 ( 36 + x ^{2} )}

= \frac{5 ( x ^{2} arctan ( \frac{x}{6} ) - 6 x + 36 arctan ( \frac{x}{6} ) )}{12 ( 36 + x ^{2} )}

Add a constant to the solution

= \frac{5 ( x ^{2} arctan ( \frac{x}{6} ) - 6 x + 36 arctan ( \frac{x}{6} ) )}{12 ( 36 + x ^{2} )} + C

\int (4 x^{\frac{1}{3}} + 1) d x

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

\int 6 x ln (9 x) d x

\int \frac{2}{2 x ^{2} + 1} d x

\int e^{- \frac{1}{2} y} d y

What is the integral of (5x^2)/((36+x^2)^2) ?
The integral of (5x^2)/((36+x^2)^2) is (5(x^2arctan(x/6)-6x+36arctan(x/6)))/(12(36+x^2))+C