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What is int (dx)/(sqrt(x^(2) + a^(2))) e...

What is `int (dx)/(sqrt(x^(2) + a^(2)))` equal to ?

A

`ln |(x + sqrt(x^(2) + a^(2)))/(a)| + c`

B

`ln |(x - sqrt(x^(2) + a^(2)))/(a)| + c`

C

`ln |(x^(2) + sqrt(x^(2) + a^(2)))/(a)| + c`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
`int (dx)/(sqrt(x^(2) + a^(2)))`
Let `x = a tan u`
`dx = a sec^(2) u du`
`= int (a sec^(2) u du)/(sqrt(a^(2) tan^(2) u + a^(2)))`
`= a int (sec^(2) u du)/(sqrt(a^(2) (1 + tan^(2) u))) rArr (a)/(a) int (sec^(2) ud u)/(sec u)`
`= int (sec^(2) u du)/(sec u) = int sec u du`
`= ln [tan (u) + sec (u)] + c " " [ :' int sec x dx = tan x]`
`= ln [(x)/(sqrt(a^(2))) + sqrt(1 + (x^(2))/(a^(2))) ] + c`
`= ln [(x)/(a) + (sqrt(a^(2) + x^(2)))/(a)] + c`
`= [(x + sqrt(x^(2) + a^(2)))/(a)] + c`
`:.` Option (a) is correct
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