What is the integral of (x^2)/(x-6) ?

The integral of (x^2)/(x-6) is ((x-6)^2)/2+12(x-6)+36ln|x-6|+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-6)

\int \frac{x ^{2}}{x - 6} d x

\frac{( x - 6 ) ^{2}}{2} + 12 (x - 6) + 36 ln ∣ x - 6 ∣ + C

Solution steps

\int \frac{x ^{2}}{x - 6} d x

Apply u-substitution

: \int \frac{( u + 6 ) ^{2}}{u} d u

= \int \frac{( u + 6 ) ^{2}}{u} d u

Expand

\frac{( u + 6 ) ^{2}}{u} : u + 12 + \frac{36}{u}

= \int u + 12 + \frac{36}{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 12 d u + \int \frac{36}{u} d u

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

\int 12 d u = 12 u

\int \frac{36}{u} d u = 36 ln ∣ u ∣

= \frac{u ^{2}}{2} + 12 u + 36 ln ∣ u ∣

Substitute back

u = x - 6

= \frac{( x - 6 ) ^{2}}{2} + 12 (x - 6) + 36 ln ∣ x - 6 ∣

Add a constant to the solution

= \frac{( x - 6 ) ^{2}}{2} + 12 (x - 6) + 36 ln ∣ x - 6 ∣ + C

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

y = \frac{2}{ln ( x )}

f (x) = \frac{1}{x ^{7}}

f (t) = 2 t^{2}

f (t) = (3 t + 1)^{\frac{2}{3}}

What is the integral of (x^2)/(x-6) ?
The integral of (x^2)/(x-6) is ((x-6)^2)/2+12(x-6)+36ln|x-6|+C