What is the integral of (169)/(x^3-13x^2) ?

The integral of (169)/(x^3-13x^2) is -ln|x|+(13)/x+ln|x-13|+C

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integral of (169)/(x^3-13x^2)

\int \frac{169}{x ^{3} - 13 x ^{2}} d x

- ln ∣ x ∣ + \frac{13}{x} + ln ∣ x - 13 ∣ + C

Solution steps

\int \frac{169}{x ^{3} - 13 x ^{2}} d x

Take the constant out:

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

= 169 \cdot \int \frac{1}{x ^{3} - 13 x ^{2}} d x

Take the partial fraction of

\frac{1}{x ^{3} - 13 x ^{2}} : - \frac{1}{169 x} - \frac{1}{13 x ^{2}} + \frac{1}{169 ( x - 13 )}

= 169 \cdot \int - \frac{1}{169 x} - \frac{1}{13 x ^{2}} + \frac{1}{169 ( x - 13 )} 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

= 169 (- \int \frac{1}{169 x} d x - \int \frac{1}{13 x ^{2}} d x + \int \frac{1}{169 ( x - 13 )} d x)

\int \frac{1}{169 x} d x = \frac{1}{169} ln ∣ x ∣

\int \frac{1}{13 x ^{2}} d x = - \frac{1}{13 x}

\int \frac{1}{169 ( x - 13 )} d x = \frac{1}{169} ln ∣ x - 13 ∣

= 169 (- \frac{1}{169} ln ∣ x ∣ - (- \frac{1}{13 x}) + \frac{1}{169} ln ∣ x - 13 ∣)

Simplify

169 (- \frac{1}{169} ln ∣ x ∣ - (- \frac{1}{13 x}) + \frac{1}{169} ln ∣ x - 13 ∣) : - ln ∣ x ∣ + \frac{13}{x} + ln ∣ x - 13 ∣

= - ln ∣ x ∣ + \frac{13}{x} + ln ∣ x - 13 ∣

Add a constant to the solution

= - ln ∣ x ∣ + \frac{13}{x} + ln ∣ x - 13 ∣ + C

x \to 2 lim (\frac{1}{x - 1})

\frac{d}{d x} ((x^{3} + 4 x)^{(x^{5} + 3 x^{2})})

\int 1 - 2 x + x^{2} d x

x \to 0 lim (\frac{ln ( 1 + x )}{x})

What is the integral of (169)/(x^3-13x^2) ?
The integral of (169)/(x^3-13x^2) is -ln|x|+(13)/x+ln|x-13|+C