What is the integral of (256)/(x^3-16x^2) ?

The integral of (256)/(x^3-16x^2) is -ln|x|+(16)/x+ln|x-16|+C

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integral of (256)/(x^3-16x^2)

\int \frac{256}{x ^{3} - 16 x ^{2}} d x

- ln ∣ x ∣ + \frac{16}{x} + ln ∣ x - 16 ∣ + C

Solution steps

\int \frac{256}{x ^{3} - 16 x ^{2}} d x

Take the constant out:

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

= 256 \cdot \int \frac{1}{x ^{3} - 16 x ^{2}} d x

Take the partial fraction of

\frac{1}{x ^{3} - 16 x ^{2}} : - \frac{1}{256 x} - \frac{1}{16 x ^{2}} + \frac{1}{256 ( x - 16 )}

= 256 \cdot \int - \frac{1}{256 x} - \frac{1}{16 x ^{2}} + \frac{1}{256 ( x - 16 )} 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

= 256 (- \int \frac{1}{256 x} d x - \int \frac{1}{16 x ^{2}} d x + \int \frac{1}{256 ( x - 16 )} d x)

\int \frac{1}{256 x} d x = \frac{1}{256} ln ∣ x ∣

\int \frac{1}{16 x ^{2}} d x = - \frac{1}{16 x}

\int \frac{1}{256 ( x - 16 )} d x = \frac{1}{256} ln ∣ x - 16 ∣

= 256 (- \frac{1}{256} ln ∣ x ∣ - (- \frac{1}{16 x}) + \frac{1}{256} ln ∣ x - 16 ∣)

Simplify

256 (- \frac{1}{256} ln ∣ x ∣ - (- \frac{1}{16 x}) + \frac{1}{256} ln ∣ x - 16 ∣) : - ln ∣ x ∣ + \frac{16}{x} + ln ∣ x - 16 ∣

= - ln ∣ x ∣ + \frac{16}{x} + ln ∣ x - 16 ∣

Add a constant to the solution

= - ln ∣ x ∣ + \frac{16}{x} + ln ∣ x - 16 ∣ + C

\frac{d}{d x} (4 x + 7)

\frac{1}{s ^{4} - 9}

\frac{1}{( s + 1 ) ( s ^{2} + 5 )}

\frac{sin ( 2 x )}{e ^{x^{2}}}

\frac{d y}{d x} = \frac{y}{x} + 3 x + 5

What is the integral of (256)/(x^3-16x^2) ?
The integral of (256)/(x^3-16x^2) is -ln|x|+(16)/x+ln|x-16|+C