∫ab(√x)/(√x+√(a+b-x))dx=.........

Correct answer: (B) (b-a)/2. Let l=∫ab(√x)/(a√x+√(a+b-x))dx ….(i) l=∫ab(√(a+b-x))/(√(a+b-x)+√(a+b-a+b-x)) l=∫ab(√(a+b-x))/(√(a+b-x)+√x) ….(ii) On adding Eqs. (i) and (ii), we get 2l=∫ab(√x+√(a+b-x))/(√x+√(a+b-x))dx ⇒2l=∫abdx=xab=b-a ⇒l=(b-a)/2

Q93 / 150 · MHT CET 2019 (Shift 2) · Mathematics

$\int_{a}^{b} \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a + b - x}} d x =.........$

A
$a + b$
Correct answer
B
$\frac{b - a}{2}$
C
$a - b$
D
$\frac{a - b}{2}$

Solution

Let

l = \int_{a}^{b} \frac{\sqrt{x}}{a \sqrt{x} + \sqrt{a + b - x}} d x

….(i)

l = \int_{a}^{b} \frac{\sqrt{a + b - x}}{\sqrt{a + b - x} + \sqrt{a + b - (a + b - x)}}

l = \int_{a}^{b} \frac{\sqrt{a + b - x}}{\sqrt{a + b - x} + \sqrt{x}}

….(ii)
On adding Eqs. (i) and (ii), we get

2 l = \int_{a}^{b} \frac{\sqrt{x} + \sqrt{a + b - x}}{\sqrt{x} + \sqrt{a + b - x}} d x

\Rightarrow 2 l = \int_{a}^{b} d x = {(x)}_{a}^{b} = b - a

\Rightarrow l = \frac{b - a}{2}

Hence, the correct answer is option: 'B'.

∫abxx+a+b-xdx=.........

Solution

Relevant Questions

$\int_{a}^{b} \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a + b - x}} d x =.........$