Question

Find the value of ${\int }_0^{a/2}\frac{adx}{(x-a)(x-2a)}$

• A. $\log \left(\frac{3}{2}\right)$
• B. $\log \left(\frac{2}{3}\right)$
• C. $(\frac{12}{7})$
• D. $\frac{13}{7}$

Answer 1

Answer: (A)

$\log \left(\frac{3}{2}\right)$

${\int }_0^{\frac{a}{2}}\frac{a}{(x-a)(x-2a)}dx$

$=-{\int }_0^{\frac{a}{2}}[\frac{1}{(x-a)}-\frac{1}{(x-2a)}]dx$

$=-{\int }_0^{\frac{a}{2}}\frac{1}{(x-a)}dx+{\int }_0^{\frac{a}{2}}\frac{1}{(x-2a)}dx$

$={|-\log (x-a\left)\right|}_0^{\frac{a}{2}}+{\left|\log \right(x-2a\left)\right|}_0^{\frac{a}{2}}$

$=-[\log (-\frac{a}{2})-\log (a\left)\right]+[\log \left(\frac{-3a}{2}\right)-\log (-2a\left)\right]$

$=-\log \left[\frac{1}{2}\right]+\log \left[\frac{3}{4}\right]$

$=-\log \left(\frac{\frac{1}{2}}{\frac{3}{4}}\right)=-\log \left(\frac{2}{3}\right)$
$=\log \frac{3}{2}$