Question

Evaluate \(\int\limits_0^5 {\frac{{\sqrt {5- x} }}{{\sqrt x + \sqrt {5- x} }}} dx\)
1. 5
2. 2.5
3. 1
4. 0

Answer 1

Correct Answer - Option 2 : 2.5

Concept:

Properties of definite integral:

\(\rm \int\limits_a^b f(x) dx\) = \(\rm \int\limits_a^b f(b+a -x) dx\)

 

Calculation:
I = \(\rm \int\limits_0^5 {\frac{{\sqrt {5- x} }}{{\sqrt x + \sqrt {5- x} }}} dx\)       ...(i)

I = \(\rm\int\limits_0^5 {\frac{{\sqrt {5 - (5+0-x)} }}{{\sqrt{5+0- x} + \sqrt {5- (5+0-x)} }}} dx\)

I = \(\rm\int\limits_0^5 {\frac{{\sqrt { x} }}{{\sqrt {5 - x} +\sqrt x }}} dx\)    ...(ii)

Adding (i) and (ii), we get

2I = \(\rm\int\limits_0^5 {\frac{{\sqrt {5- x}+\sqrt x }}{{\sqrt x + \sqrt {5- x} }}} dx\)

2I = \(\rm\int\limits_0^5dx\)

2I = \(\rm\left[x\right]_0^5\)

2I = 5 - 0 = 5

I = 2.5