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

The integral of (x^3)/(x^3-7x+6) is x-1/4 ln|x-1|+8/5 ln|x-2|-27/20 ln|x+3|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

\int \frac{x ^{3}}{x ^{3} - 7 x + 6} d x

x - \frac{1}{4} ln ∣ x - 1 ∣ + \frac{8}{5} ln ∣ x - 2 ∣ - \frac{27}{20} ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{x ^{3}}{x ^{3} - 7 x + 6} d x

Take the partial fraction of

\frac{x ^{3}}{x ^{3} - 7 x + 6} : 1 - \frac{1}{4 ( x - 1 )} + \frac{8}{5 ( x - 2 )} - \frac{27}{20 ( x + 3 )}

= \int 1 - \frac{1}{4 ( x - 1 )} + \frac{8}{5 ( x - 2 )} - \frac{27}{20 ( x + 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 1 d x - \int \frac{1}{4 ( x - 1 )} d x + \int \frac{8}{5 ( x - 2 )} d x - \int \frac{27}{20 ( x + 3 )} d x

\int 1 d x = x

\int \frac{1}{4 ( x - 1 )} d x = \frac{1}{4} ln ∣ x - 1 ∣

\int \frac{8}{5 ( x - 2 )} d x = \frac{8}{5} ln ∣ x - 2 ∣

\int \frac{27}{20 ( x + 3 )} d x = \frac{27}{20} ln ∣ x + 3 ∣

= x - \frac{1}{4} ln ∣ x - 1 ∣ + \frac{8}{5} ln ∣ x - 2 ∣ - \frac{27}{20} ln ∣ x + 3 ∣

Add a constant to the solution

= x - \frac{1}{4} ln ∣ x - 1 ∣ + \frac{8}{5} ln ∣ x - 2 ∣ - \frac{27}{20} ln ∣ x + 3 ∣ + C

\frac{\partial}{\partial y} (sin (3 x + 4 y))

\int \frac{cos ( t )}{1 + sin ( t )} d t

\frac{\partial}{\partial x} (x^{2} ln (x))

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

f (x) = ln (cos (2 x))

What is the integral of (x^3)/(x^3-7x+6) ?
The integral of (x^3)/(x^3-7x+6) is x-1/4 ln|x-1|+8/5 ln|x-2|-27/20 ln|x+3|+C