What is the integral of x^2*ln((x-1)/x) ?

The integral of x^2*ln((x-1)/x) is 1/3 x^3ln((x-1)/x)-1/6 (x^2+2x+2ln|x-1|-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^2*ln((x-1)/x)

\int x^{2} \cdot ln (\frac{x - 1}{x}) d x

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

Solution steps

\int x^{2} ln (\frac{x - 1}{x}) d x

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

\int \frac{z ^{2}}{z ^{2} + a ^{2}} d z

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

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

\int - π sin (π) θ d θ

\int x^{4} (1 - x)^{4} d x

What is the integral of x^2*ln((x-1)/x) ?
The integral of x^2*ln((x-1)/x) is 1/3 x^3ln((x-1)/x)-1/6 (x^2+2x+2ln|x-1|-3)+C