What is the integral of (36)/(x^3-6x^2) ?

The integral of (36)/(x^3-6x^2) is -ln|x|+6/x+ln|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 (36)/(x^3-6x^2)

\int \frac{36}{x ^{3} - 6 x ^{2}} d x

- ln ∣ x ∣ + \frac{6}{x} + ln ∣ x - 6 ∣ + C

Solution steps

\int \frac{36}{x ^{3} - 6 x ^{2}} d x

Take the constant out:

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

= 36 \cdot \int \frac{1}{x ^{3} - 6 x ^{2}} d x

Take the partial fraction of

\frac{1}{x ^{3} - 6 x ^{2}} : - \frac{1}{36 x} - \frac{1}{6 x ^{2}} + \frac{1}{36 ( x - 6 )}

= 36 \cdot \int - \frac{1}{36 x} - \frac{1}{6 x ^{2}} + \frac{1}{36 ( x - 6 )} 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

= 36 (- \int \frac{1}{36 x} d x - \int \frac{1}{6 x ^{2}} d x + \int \frac{1}{36 ( x - 6 )} d x)

\int \frac{1}{36 x} d x = \frac{1}{36} ln ∣ x ∣

\int \frac{1}{6 x ^{2}} d x = - \frac{1}{6 x}

\int \frac{1}{36 ( x - 6 )} d x = \frac{1}{36} ln ∣ x - 6 ∣

= 36 (- \frac{1}{36} ln ∣ x ∣ - (- \frac{1}{6 x}) + \frac{1}{36} ln ∣ x - 6 ∣)

Simplify

36 (- \frac{1}{36} ln ∣ x ∣ - (- \frac{1}{6 x}) + \frac{1}{36} ln ∣ x - 6 ∣) : - ln ∣ x ∣ + \frac{6}{x} + ln ∣ x - 6 ∣

= - ln ∣ x ∣ + \frac{6}{x} + ln ∣ x - 6 ∣

Add a constant to the solution

= - ln ∣ x ∣ + \frac{6}{x} + ln ∣ x - 6 ∣ + C

\int e^{9 x + 7} d x

\frac{d}{d x} (e^{3 x + 9})

(x^{3} - 9 x)^{10} (3)

(\frac{2 x}{x ^{2} + 1})^{^{'}}

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

What is the integral of (36)/(x^3-6x^2) ?
The integral of (36)/(x^3-6x^2) is -ln|x|+6/x+ln|x-6|+C