. What is the exact value of ? Give your answer as a fraction or whole number
I know i have to split into partial fractions, but not sure how to go from there
That's the correct partial fraction decomposition. In the second line, instead of an integral sign, you want -ln(x+1)+ln(x+2) evaluated from 0 to 1, so it's -ln(1+1)+ln(1+2)+ln(0+1)-ln(0+2). Now you can combine the logarithms to get k.
- Hollywood
Let me just touch it up a bit:
$\displaystyle \int^1_0 \frac{-1}{(x+1)(x+2)} \ dx$
$\displaystyle = \int^1_0 \left( \frac{-1}{x+1} + \frac{1}{x+2} \right) dx$
$\displaystyle = \int^1_0 \frac{-1}{x+1} \ dx + \int^1_0 \frac{1}{x+2} \ dx$
$\displaystyle = \left\ -ln(x+1)\right]_0^1 + \left\ ln(x+2)\right]_0^1 $
And from there:
Spoiler:
My spoiler is spoiled!
From $\displaystyle -ln(2) + ln(1) + ln(3) - ln(2)$ to $\displaystyle ln(1) + ln(3)$? Not my best bit of calculation.
It should be
$\displaystyle -ln(2) + ln(1) + ln(3) - ln(2) = ln(3)-2ln(2) = ln(3) - ln(2^2) = ln(3/4)$.
Thanks for catching my error.