I did an integral and the answer on wolfram is $\frac{1}{5} ln{\frac{3}{2}} + ln{2}$ and it's equal to 0.77424 which is == to my answer which is $\frac{3}{5}(ln3 - ln1) + \frac{2}{5}(ln8-ln6)$
Why are these 2 answers the same? CAn someone help me simplify?
I can get to $\frac{3}{5} ln3 + \frac{2}{5} ln(\frac{4}{3})$
\begin{align} \frac35 \ln 3 + \frac25 \ln \left( \frac43\right)&=\frac35 \ln 3 + \frac25 \ln \left(4\right)- \frac25 \ln \left(3\right)\\ &=\frac15 \ln 3 + \frac25 \ln 4 \\ &=\frac15 \ln 3 + \frac25 \ln 2^2 \\ &=\frac15 \ln 3 + \frac45 \ln 2 \\ &=\frac15 \ln 3 +\left( 1-\frac15\right) \ln 2 \\ &=\frac15 \left(\ln 3 -\ln 2\right) + \ln 2\\ &=\frac15 \ln \frac32 + \ln 2 \end{align}
By log properties $ \ln a + \ln b= ln (ab)$, same with $ \ln a - \ln b= ln a + (-ln b)= ln a + ln b^- = ln (ab^-)= ln$ $a \over {b}$, thus
$3\over{5}$$(ln3−ln1)= $$3\over{5}$$ln3$, and $2\over{5}$$(ln8−ln6)= $$2\over{5}$$ln$$8\over6$ = $2\over 5$$ln$$4\over 3$
