Question

What is the integration of `int1/(1-t)dt - int 1/(2-t)dt` ?

`int1/(1-t)dt - int 1/(2-t)dt`

Answer 1

`\ln|\frac{t-2}{t-1}|+C`

Explanation:

Given that

`\int \frac{1}{1-t}\ dt-\int \frac{1}{2-t}\ dt`

`=-\int \frac{(-1)}{1-t}\ dt+\int \frac{(-1)}{2-t}\ dt`

`=-\int \frac{d(1-t)}{1-t}+\int \frac{d(2-t)}{2-t}`

`=-\ln|1-t|+\ln|2-t|+C`

`=\ln|\frac{2-t}{1-t}|+C`

`=\ln|\frac{t-2}{t-1}|+C`