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

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

integral

\frac{x ^{2}}{( x - 1 )}

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

Solution steps

\int \frac{x ^{2}}{x - 1} d x

Apply u-substitution

: \int \frac{( u + 1 ) ^{2}}{u} d u

= \int \frac{( u + 1 ) ^{2}}{u} d u

Expand

\frac{( u + 1 ) ^{2}}{u} : u + 2 + \frac{1}{u}

= \int u + 2 + \frac{1}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u d u + \int 2 d u + \int \frac{1}{u} d u

\int u d u = \frac{u ^{2}}{2}

\int 2 d u = 2 u

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{u ^{2}}{2} + 2 u + ln ∣ u ∣

Substitute back

u = x - 1

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

Add a constant to the solution

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

x^{2} a^{2} + x^{2}

\frac{sin ^{3} ( x )}{cos ( x )}

sin (5 x + 3)

\frac{x ^{2}}{x ^{2} + 5}

\frac{( e ^{x} ) ^{2}}{x}

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