Question

The value of integral $\underset{20}{\overset{30}{\int }}\frac{\ln x}{\ln x+\ln (50-x)}dx$ is

• A. $5$
• B. $10$
• C. $15$
• D. $-5$

Answer 1

The correct option is A $5$

Let $I=\underset{20}{\overset{30}{\int }}\frac{\ln x}{\ln x+\ln (50-x)}dx\cdots \left(i\right)$

Using property
$\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{a}{\overset{b}{\int }}f(a+b-x)dx$
We get, $I=\underset{20}{\overset{30}{\int }}\frac{\ln (50-x)}{\ln (50-x)+\ln x}dx\cdots \left(ii\right)$
On adding $\left(i\right)$ and $\left(ii\right),$ we get
$2I=\underset{20}{\overset{30}{\int }}1dx=10$

$\Rightarrow I=5$