What is the integral of x/((x-1)(x-2)) ?

The integral of x/((x-1)(x-2)) is -ln|x-1|+2ln|x-2|+C

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integral of x/((x-1)(x-2))

\int \frac{x}{( x - 1 ) ( x - 2 )} d x

- ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{x}{( x - 1 ) ( x - 2 )} d x

Take the partial fraction of

\frac{x}{( x - 1 ) ( x - 2 )} : - \frac{1}{x - 1} + \frac{2}{x - 2}

= \int - \frac{1}{x - 1} + \frac{2}{x - 2} 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

= - \int \frac{1}{x - 1} d x + \int \frac{2}{x - 2} d x

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

\int \frac{2}{x - 2} d x = 2 ln ∣ x - 2 ∣

= - ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣

Add a constant to the solution

= - ln ∣ x - 1 ∣ + 2 ln ∣ x - 2 ∣ + C

\int y arctan (2 y) d y

6 y^{^{''}} + y = 0

y^{^{'}} - \frac{6}{x} y = \frac{y ^{3}}{x ^{9}}

f (x) = 2 x^{2} - x, (1, 1)

\frac{\partial}{\partial x} (c)

What is the integral of x/((x-1)(x-2)) ?
The integral of x/((x-1)(x-2)) is -ln|x-1|+2ln|x-2|+C