UPSC NDA Math Model Paper 3

Here, we have to find the value of the integral

π 2

∫

0

√tan‌x

1+√tan‌x

Let I=

π 2

∫

0

√tan‌x

1+√tan‌x

dx ......(1)
As we know that,

b

∫

a

f(x)dx=

b

∫

a

f(a+b−x)dx
I=

π 2

∫

0

√cot‌x

1+√cot‌x

dx ......(2)
Adding equation (1) and (2), we get
⇒2I=

π 2

∫

0

[

√tan‌x

1+√tan‌x

+

√cot‌x

1+√cot‌x

]dx
⇒2I=

π 2

∫

0

dx=

π

2

⇒I=

π

4

Hence, option B is the correct answer.