_0^ 3 f(x) f(x)+f ( -3 x 3 ) d x= - Vidyadip

MCQ

$\int_0^{\frac{\alpha}{3}} \frac{f(x)}{f(x)+f\left(\frac{\alpha-3 x}{3}\right)} d x=$

A
$\frac{2 \alpha}{3}$
B
$\frac{\alpha}{2}$
C
$\frac{\alpha}{3}$
✓
$\frac{\alpha}{6}$

✓

Answer

Correct option: D.

$\frac{\alpha}{6}$

(D)
Let $I=\int_0^{\frac{\alpha}{3}} \frac{f(x)}{f(x)+f\left(\frac{\alpha-3 x}{3}\right)} d x$
$\Rightarrow I =\int_0^{\frac{\alpha}{3}} \frac{ f (x)}{ f (x)+ f \left(\frac{\alpha}{3}-x\right)} d x$
$\int_0^{ a } \frac{ f (x)}{ f (x)+ f ( a -x)} d x=\frac{ a }{2}$
$=\frac{\frac{\alpha}{3}}{2} $
$=\frac{\alpha}{6}$

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