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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

- 4 (\frac{x ^{4}}{2 x ^{3} + x ^{2} - 2 x - 1} + \frac{1}{12} ln ∣ 2 x + 1 ∣ + \frac{1}{12 ( 2 x + 1 )} - \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2 ( x + 1 )} + \frac{1}{6} ln ∣ x - 1 ∣ - \frac{1}{6 ( x - 1 )}) + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

\int \frac{x ( x ^{4} - 4 x ^{3} - 3 x ^{2} )}{( 2 x ^{3} + x ^{2} - 2 x - 1 ) ^{2}} d x = - \frac{1}{12} ln ∣ 2 x + 1 ∣ - \frac{1}{12 ( 2 x + 1 )} + \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2 ( x + 1 )} - \frac{1}{6} ln ∣ x - 1 ∣ + \frac{1}{6 ( x - 1 )}

= - 4 (\frac{x ^{4}}{2 x ^{3} + x ^{2} - 2 x - 1} - (- \frac{1}{12} ln ∣ 2 x + 1 ∣ - \frac{1}{12 ( 2 x + 1 )} + \frac{1}{2} ln ∣ x + 1 ∣ + \frac{1}{2 ( x + 1 )} - \frac{1}{6} ln ∣ x - 1 ∣ + \frac{1}{6 ( x - 1 )}))

Simplify

= - 4 (\frac{x ^{4}}{2 x ^{3} + x ^{2} - 2 x - 1} + \frac{1}{12} ln ∣ 2 x + 1 ∣ + \frac{1}{12 ( 2 x + 1 )} - \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2 ( x + 1 )} + \frac{1}{6} ln ∣ x - 1 ∣ - \frac{1}{6 ( x - 1 )})

Add a constant to the solution

= - 4 (\frac{x ^{4}}{2 x ^{3} + x ^{2} - 2 x - 1} + \frac{1}{12} ln ∣ 2 x + 1 ∣ + \frac{1}{12 ( 2 x + 1 )} - \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2 ( x + 1 )} + \frac{1}{6} ln ∣ x - 1 ∣ - \frac{1}{6 ( x - 1 )}) + C

x \to 1 lim (5 x^{4} - 5 x^{2} + 10)

f^{^{'}} (x) = f (x)^{4} \cdot (4 x^{2} + x)

\int \frac{6 x - 7}{( 2 x + 1 ) ( 4 x ^{2} + 1 )} d x

s^{2} + 4 s + 13

y^{^{'}} = 3 x + x y

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