What is the integral of ((3x^4+2x^2))/((x^2+1)) ?

The integral of ((3x^4+2x^2))/((x^2+1)) is x^3-x+arctan(x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of ((3x^4+2x^2))/((x^2+1))

\int \frac{( 3 x ^{4} + 2 x ^{2} )}{( x ^{2} + 1 )} d x

x^{3} - x + arctan (x) + C

Solution steps

\int \frac{3 x ^{4} + 2 x ^{2}}{x ^{2} + 1} d x

Expand

\frac{3 x ^{4} + 2 x ^{2}}{x ^{2} + 1} : \frac{3 x ^{4}}{x ^{2} + 1} + \frac{2 x ^{2}}{x ^{2} + 1}

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{3 x ^{4}}{x ^{2} + 1} d x + \int \frac{2 x ^{2}}{x ^{2} + 1} d x

\int \frac{3 x ^{4}}{x ^{2} + 1} d x = x^{3} - 3 x + 3 arctan (x)

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

= x^{3} - 3 x + 3 arctan (x) + 2 (- arctan (x) + x)

Simplify

x^{3} - 3 x + 3 arctan (x) + 2 (- arctan (x) + x) : x^{3} - x + arctan (x)

= x^{3} - x + arctan (x)

Add a constant to the solution

= x^{3} - x + arctan (x) + C

\int \frac{x ^{2} + 6 x + 4}{x ^{3} + 9 x ^{2} + 12 x + 9} d x

\int 3 x^{3} - 2 \cdot (6 x^{2}) d x

\int 2 x^{\frac{4}{5}} d x

\int - x^{2} - 2 x + 2 d x

\int 6 x sin (3 x) d x

What is the integral of ((3x^4+2x^2))/((x^2+1)) ?
The integral of ((3x^4+2x^2))/((x^2+1)) is x^3-x+arctan(x)+C