What is the integral of (400)/(x^3-20x^2) ?

The integral of (400)/(x^3-20x^2) is -ln|x|+(20)/x+ln|x-20|+C

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integral of (400)/(x^3-20x^2)

\int \frac{400}{x ^{3} - 20 x ^{2}} d x

- ln ∣ x ∣ + \frac{20}{x} + ln ∣ x - 20 ∣ + C

Solution steps

\int \frac{400}{x ^{3} - 20 x ^{2}} d x

Take the constant out:

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

= 400 \cdot \int \frac{1}{x ^{3} - 20 x ^{2}} d x

Take the partial fraction of

\frac{1}{x ^{3} - 20 x ^{2}} : - \frac{1}{400 x} - \frac{1}{20 x ^{2}} + \frac{1}{400 ( x - 20 )}

= 400 \cdot \int - \frac{1}{400 x} - \frac{1}{20 x ^{2}} + \frac{1}{400 ( x - 20 )} 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

= 400 (- \int \frac{1}{400 x} d x - \int \frac{1}{20 x ^{2}} d x + \int \frac{1}{400 ( x - 20 )} d x)

\int \frac{1}{400 x} d x = \frac{1}{400} ln ∣ x ∣

\int \frac{1}{20 x ^{2}} d x = - \frac{1}{20 x}

\int \frac{1}{400 ( x - 20 )} d x = \frac{1}{400} ln ∣ x - 20 ∣

= 400 (- \frac{1}{400} ln ∣ x ∣ - (- \frac{1}{20 x}) + \frac{1}{400} ln ∣ x - 20 ∣)

Simplify

400 (- \frac{1}{400} ln ∣ x ∣ - (- \frac{1}{20 x}) + \frac{1}{400} ln ∣ x - 20 ∣) : - ln ∣ x ∣ + \frac{20}{x} + ln ∣ x - 20 ∣

= - ln ∣ x ∣ + \frac{20}{x} + ln ∣ x - 20 ∣

Add a constant to the solution

= - ln ∣ x ∣ + \frac{20}{x} + ln ∣ x - 20 ∣ + C

\int_{4}^{16} \frac{1}{1 - x} d x

f (x) = 4 x^{2} + 4 x, at x = - 2

\int e^{- \frac{1}{4} x^{2}} d x

\frac{d}{d x} (1 - (1 - \frac{x}{40})^{3})

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

What is the integral of (400)/(x^3-20x^2) ?
The integral of (400)/(x^3-20x^2) is -ln|x|+(20)/x+ln|x-20|+C