What is the integral of (2x^3+5x)/((x^2+3)^2) ?

The integral of (2x^3+5x)/((x^2+3)^2) is ln|x^2+3|+3/(x^2+3)-5/(2(x^2+3))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (2x^3+5x)/((x^2+3)^2)

\int \frac{2 x ^{3} + 5 x}{( x ^{2} + 3 ) ^{2}} d x

ln x^{2} + 3 + \frac{3}{x ^{2} + 3} - \frac{5}{2 ( x ^{2} + 3 )} + C

Solution steps

\int \frac{2 x ^{3} + 5 x}{( x ^{2} + 3 ) ^{2}} d x

Expand

\frac{2 x ^{3} + 5 x}{( x ^{2} + 3 ) ^{2}} : \frac{2 x ^{3}}{( x ^{2} + 3 ) ^{2}} + \frac{5 x}{( x ^{2} + 3 ) ^{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{2 x ^{3}}{( x ^{2} + 3 ) ^{2}} d x + \int \frac{5 x}{( x ^{2} + 3 ) ^{2}} d x

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

\int \frac{5 x}{( x ^{2} + 3 ) ^{2}} d x = - \frac{5}{2 ( x ^{2} + 3 )}

= ln x^{2} + 3 + \frac{3}{x ^{2} + 3} - \frac{5}{2 ( x ^{2} + 3 )}

Add a constant to the solution

= ln x^{2} + 3 + \frac{3}{x ^{2} + 3} - \frac{5}{2 ( x ^{2} + 3 )} + C

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

\int (csc^{2} (x) - 3 csc (x) cot (x)) d x

\int \frac{- sin ( 2 x ) tan ( 2 x )}{2} d x

\int \frac{5 x + 8}{x ^{2} + 2 x - 8} d x

\int x^{3} + y^{3} d x

What is the integral of (2x^3+5x)/((x^2+3)^2) ?
The integral of (2x^3+5x)/((x^2+3)^2) is ln|x^2+3|+3/(x^2+3)-5/(2(x^2+3))+C