What is the integral of (289)/(x^3-17x^2) ?

The integral of (289)/(x^3-17x^2) is -ln|x|+(17)/x+ln|x-17|+C

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integral of (289)/(x^3-17x^2)

\int \frac{289}{x ^{3} - 17 x ^{2}} d x

- ln ∣ x ∣ + \frac{17}{x} + ln ∣ x - 17 ∣ + C

Solution steps

\int \frac{289}{x ^{3} - 17 x ^{2}} d x

Take the constant out:

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

= 289 \cdot \int \frac{1}{x ^{3} - 17 x ^{2}} d x

Take the partial fraction of

\frac{1}{x ^{3} - 17 x ^{2}} : - \frac{1}{289 x} - \frac{1}{17 x ^{2}} + \frac{1}{289 ( x - 17 )}

= 289 \cdot \int - \frac{1}{289 x} - \frac{1}{17 x ^{2}} + \frac{1}{289 ( x - 17 )} 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

= 289 (- \int \frac{1}{289 x} d x - \int \frac{1}{17 x ^{2}} d x + \int \frac{1}{289 ( x - 17 )} d x)

\int \frac{1}{289 x} d x = \frac{1}{289} ln ∣ x ∣

\int \frac{1}{17 x ^{2}} d x = - \frac{1}{17 x}

\int \frac{1}{289 ( x - 17 )} d x = \frac{1}{289} ln ∣ x - 17 ∣

= 289 (- \frac{1}{289} ln ∣ x ∣ - (- \frac{1}{17 x}) + \frac{1}{289} ln ∣ x - 17 ∣)

Simplify

289 (- \frac{1}{289} ln ∣ x ∣ - (- \frac{1}{17 x}) + \frac{1}{289} ln ∣ x - 17 ∣) : - ln ∣ x ∣ + \frac{17}{x} + ln ∣ x - 17 ∣

= - ln ∣ x ∣ + \frac{17}{x} + ln ∣ x - 17 ∣

Add a constant to the solution

= - ln ∣ x ∣ + \frac{17}{x} + ln ∣ x - 17 ∣ + C

\int (1 - t) d t

x \to 0 + lim (I n x)

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

\int_{3}^{x^{3}} t^{3} d t

\frac{1}{x}

What is the integral of (289)/(x^3-17x^2) ?
The integral of (289)/(x^3-17x^2) is -ln|x|+(17)/x+ln|x-17|+C