What is the integral of (7x^2)/((225+x^2)^2) ?

The integral of (7x^2)/((225+x^2)^2) is (7(x^2arctan(x/(15))-15x+225arctan(x/(15))))/(30(225+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 (7x^2)/((225+x^2)^2)

\int \frac{7 x ^{2}}{( 225 + x ^{2} ) ^{2}} d x

\frac{7 ( x ^{2} arctan ( \frac{x}{15} ) - 15 x + 225 arctan ( \frac{x}{15} ) )}{30 ( 225 + x ^{2} )} + C

Solution steps

\int \frac{7 x ^{2}}{( 225 + x ^{2} ) ^{2}} d x

Take the constant out:

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

= 7 \cdot \int \frac{x ^{2}}{( 225 + x ^{2} ) ^{2}} d x

Apply Integral Substitution

= 7 \cdot \int \frac{u ^{2}}{15 ( u ^{2} + 1 ) ^{2}} d u

Take the constant out:

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

= 7 \cdot \frac{1}{15} \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}}

= 7 \cdot \frac{1}{15} \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

= 7 \cdot \frac{1}{15} (\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}})

= 7 \cdot \frac{1}{15} (arctan (u) - \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}}))

Substitute back

u = \frac{x}{15}

= 7 \cdot \frac{1}{15} (arctan (\frac{x}{15}) - \frac{1}{2} (arctan (\frac{x}{15}) + \frac{\frac{x}{15}}{1 + ( \frac{x}{15} ) ^{2}}))

Simplify

7 \cdot \frac{1}{15} (arctan (\frac{x}{15}) - \frac{1}{2} (arctan (\frac{x}{15}) + \frac{\frac{x}{15}}{1 + ( \frac{x}{15} ) ^{2}})) : \frac{7 ( x ^{2} arctan ( \frac{x}{15} ) - 15 x + 225 arctan ( \frac{x}{15} ) )}{30 ( 225 + x ^{2} )}

= \frac{7 ( x ^{2} arctan ( \frac{x}{15} ) - 15 x + 225 arctan ( \frac{x}{15} ) )}{30 ( 225 + x ^{2} )}

Add a constant to the solution

= \frac{7 ( x ^{2} arctan ( \frac{x}{15} ) - 15 x + 225 arctan ( \frac{x}{15} ) )}{30 ( 225 + x ^{2} )} + C

x \to 0 + lim ((5 x^{2}) (ln (x)))

y = \frac{x ^{2} - 1}{x ^{2} + x + 1}, at x = 1

\int 4 - 4 sin (x) + sin^{2} (x) d x

\int 64 - 3 x^{2} d x

\frac{\partial}{\partial x} (x^{2} e^{2})

What is the integral of (7x^2)/((225+x^2)^2) ?
The integral of (7x^2)/((225+x^2)^2) is (7(x^2arctan(x/(15))-15x+225arctan(x/(15))))/(30(225+x^2))+C