What is the integral of-((2x-9))/((x^2+3)^3) ?

The integral of-((2x-9))/((x^2+3)^3) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of-((2x-9))/((x^2+3)^3)

\int - \frac{( 2 x - 9 )}{( x ^{2} + 3 ) ^{3}} d x

\frac{1}{2 ( x ^{2} + 3 ) ^{2}} + \frac{3 ( x ^{4} arctan ( \frac{x}{3} ) + 3 x ^{3} + 6 x ^{2} arctan ( \frac{x}{3} ) + 5 3 x + 9 arctan ( \frac{x}{3} ) )}{8 ( x ^{2} + 3 ) ^{2}} + C

Solution steps

\int - \frac{2 x - 9}{( x ^{2} + 3 ) ^{3}} d x

Take the constant out:

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

= - \int \frac{2 x - 9}{( x ^{2} + 3 ) ^{3}} d x

Expand

\frac{2 x - 9}{( x ^{2} + 3 ) ^{3}} : \frac{2 x}{( x ^{2} + 3 ) ^{3}} - \frac{9}{( x ^{2} + 3 ) ^{3}}

= - \int \frac{2 x}{( x ^{2} + 3 ) ^{3}} - \frac{9}{( x ^{2} + 3 ) ^{3}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (\int \frac{2 x}{( x ^{2} + 3 ) ^{3}} d x - \int \frac{9}{( x ^{2} + 3 ) ^{3}} d x)

\int \frac{2 x}{( x ^{2} + 3 ) ^{3}} d x = - \frac{1}{2 ( x ^{2} + 3 ) ^{2}}

\int \frac{9}{( x ^{2} + 3 ) ^{3}} d x = \frac{3 ( x ^{4} arctan ( \frac{x}{3} ) + 3 x ^{3} + 6 x ^{2} arctan ( \frac{x}{3} ) + 5 3 x + 9 arctan ( \frac{x}{3} ) )}{8 ( x ^{2} + 3 ) ^{2}}

= - - \frac{1}{2 ( x ^{2} + 3 ) ^{2}} - \frac{3 ( x ^{4} arctan ( \frac{x}{3} ) + 3 x ^{3} + 6 x ^{2} arctan ( \frac{x}{3} ) + 5 3 x + 9 arctan ( \frac{x}{3} ) )}{8 ( x ^{2} + 3 ) ^{2}}

Simplify

= \frac{1}{2 ( x ^{2} + 3 ) ^{2}} + \frac{3 ( x ^{4} arctan ( \frac{x}{3} ) + 3 x ^{3} + 6 x ^{2} arctan ( \frac{x}{3} ) + 5 3 x + 9 arctan ( \frac{x}{3} ) )}{8 ( x ^{2} + 3 ) ^{2}}

Add a constant to the solution

= \frac{1}{2 ( x ^{2} + 3 ) ^{2}} + \frac{3 ( x ^{4} arctan ( \frac{x}{3} ) + 3 x ^{3} + 6 x ^{2} arctan ( \frac{x}{3} ) + 5 3 x + 9 arctan ( \frac{x}{3} ) )}{8 ( x ^{2} + 3 ) ^{2}} + C

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

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

\int cot^{3} (a) csc^{3} (a) d a

\int \frac{e ^{3 x} + 1}{5 e ^{2 x}} d x

e^{7 x}

What is the integral of-((2x-9))/((x^2+3)^3) ?
The integral of-((2x-9))/((x^2+3)^3) is