What is the integral of (441)/(x^3-21x^2) ?

The integral of (441)/(x^3-21x^2) is -ln|x|+(21)/x+ln|x-21|+C

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integral of (441)/(x^3-21x^2)

\int \frac{441}{x ^{3} - 21 x ^{2}} d x

- ln ∣ x ∣ + \frac{21}{x} + ln ∣ x - 21 ∣ + C

Solution steps

\int \frac{441}{x ^{3} - 21 x ^{2}} d x

Take the constant out:

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

= 441 \cdot \int \frac{1}{x ^{3} - 21 x ^{2}} d x

Take the partial fraction of

\frac{1}{x ^{3} - 21 x ^{2}} : - \frac{1}{441 x} - \frac{1}{21 x ^{2}} + \frac{1}{441 ( x - 21 )}

= 441 \cdot \int - \frac{1}{441 x} - \frac{1}{21 x ^{2}} + \frac{1}{441 ( x - 21 )} 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

= 441 (- \int \frac{1}{441 x} d x - \int \frac{1}{21 x ^{2}} d x + \int \frac{1}{441 ( x - 21 )} d x)

\int \frac{1}{441 x} d x = \frac{1}{441} ln ∣ x ∣

\int \frac{1}{21 x ^{2}} d x = - \frac{1}{21 x}

\int \frac{1}{441 ( x - 21 )} d x = \frac{1}{441} ln ∣ x - 21 ∣

= 441 (- \frac{1}{441} ln ∣ x ∣ - (- \frac{1}{21 x}) + \frac{1}{441} ln ∣ x - 21 ∣)

Simplify

441 (- \frac{1}{441} ln ∣ x ∣ - (- \frac{1}{21 x}) + \frac{1}{441} ln ∣ x - 21 ∣) : - ln ∣ x ∣ + \frac{21}{x} + ln ∣ x - 21 ∣

= - ln ∣ x ∣ + \frac{21}{x} + ln ∣ x - 21 ∣

Add a constant to the solution

= - ln ∣ x ∣ + \frac{21}{x} + ln ∣ x - 21 ∣ + C

\int \frac{x ^{2}}{( 25 + x ^{2} ) ^{2}} d x

5 x y \frac{d y}{d x} = 1

y^{^{''}} - 3 y^{^{'}} - 12 y = 0

\frac{4 e ^{(- 2 s)}}{( s ^{2} - 4 s + 3 )}

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

What is the integral of (441)/(x^3-21x^2) ?
The integral of (441)/(x^3-21x^2) is -ln|x|+(21)/x+ln|x-21|+C