What is the integral of (6x^3+7x)/((x^2+2)^2) ?

The integral of (6x^3+7x)/((x^2+2)^2) is 3(ln|x^2+2|+2/(x^2+2))-7/(2(x^2+2))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (6x^3+7x)/((x^2+2)^2)

\int \frac{6 x ^{3} + 7 x}{( x ^{2} + 2 ) ^{2}} d x

3 (ln x^{2} + 2 + \frac{2}{x ^{2} + 2}) - \frac{7}{2 ( x ^{2} + 2 )} + C

Solution steps

\int \frac{6 x ^{3} + 7 x}{( x ^{2} + 2 ) ^{2}} d x

Expand

\frac{6 x ^{3} + 7 x}{( x ^{2} + 2 ) ^{2}} : \frac{6 x ^{3}}{( x ^{2} + 2 ) ^{2}} + \frac{7 x}{( x ^{2} + 2 ) ^{2}}

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{6 x ^{3}}{( x ^{2} + 2 ) ^{2}} d x + \int \frac{7 x}{( x ^{2} + 2 ) ^{2}} d x

\int \frac{6 x ^{3}}{( x ^{2} + 2 ) ^{2}} d x = 3 (ln x^{2} + 2 + \frac{2}{x ^{2} + 2})

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

= 3 (ln x^{2} + 2 + \frac{2}{x ^{2} + 2}) - \frac{7}{2 ( x ^{2} + 2 )}

Add a constant to the solution

= 3 (ln x^{2} + 2 + \frac{2}{x ^{2} + 2}) - \frac{7}{2 ( x ^{2} + 2 )} + C

\frac{2}{( s + 1 ) ^{3}}

\frac{d}{d x} (- 4 e^{- 2.5 x})

\int sin^{7} (x) d x

\frac{d}{d x} (ln (x + 4))

\int \frac{x ^{2}}{( 2 x + 1 ) ^{5}} d x

What is the integral of (6x^3+7x)/((x^2+2)^2) ?
The integral of (6x^3+7x)/((x^2+2)^2) is 3(ln|x^2+2|+2/(x^2+2))-7/(2(x^2+2))+C