What is the integral of (100)/(x^3-10x^2) ?

The integral of (100)/(x^3-10x^2) is -ln|x|+(10)/x+ln|x-10|+C

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integral of (100)/(x^3-10x^2)

\int \frac{100}{x ^{3} - 10 x ^{2}} d x

- ln ∣ x ∣ + \frac{10}{x} + ln ∣ x - 10 ∣ + C

Solution steps

\int \frac{100}{x ^{3} - 10 x ^{2}} d x

Take the constant out:

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

= 100 \cdot \int \frac{1}{x ^{3} - 10 x ^{2}} d x

Take the partial fraction of

\frac{1}{x ^{3} - 10 x ^{2}} : - \frac{1}{100 x} - \frac{1}{10 x ^{2}} + \frac{1}{100 ( x - 10 )}

= 100 \cdot \int - \frac{1}{100 x} - \frac{1}{10 x ^{2}} + \frac{1}{100 ( x - 10 )} 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

= 100 (- \int \frac{1}{100 x} d x - \int \frac{1}{10 x ^{2}} d x + \int \frac{1}{100 ( x - 10 )} d x)

\int \frac{1}{100 x} d x = \frac{1}{100} ln ∣ x ∣

\int \frac{1}{10 x ^{2}} d x = - \frac{1}{10 x}

\int \frac{1}{100 ( x - 10 )} d x = \frac{1}{100} ln ∣ x - 10 ∣

= 100 (- \frac{1}{100} ln ∣ x ∣ - (- \frac{1}{10 x}) + \frac{1}{100} ln ∣ x - 10 ∣)

Simplify

100 (- \frac{1}{100} ln ∣ x ∣ - (- \frac{1}{10 x}) + \frac{1}{100} ln ∣ x - 10 ∣) : - ln ∣ x ∣ + \frac{10}{x} + ln ∣ x - 10 ∣

= - ln ∣ x ∣ + \frac{10}{x} + ln ∣ x - 10 ∣

Add a constant to the solution

= - ln ∣ x ∣ + \frac{10}{x} + ln ∣ x - 10 ∣ + C

\int_{- 6}^{6} π (36 - x^{2}) d x

\frac{d y}{d x} + \frac{1}{x} y = 6 x^{2}

f (x) = \frac{1}{( 1 + x ^{2} )}, at x = - 1

x \to - 10 lim ((+) f (x))

\frac{\partial}{\partial y} (e^{13 x y})

What is the integral of (100)/(x^3-10x^2) ?
The integral of (100)/(x^3-10x^2) is -ln|x|+(10)/x+ln|x-10|+C