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If int(dx)/((x+100)sqrt(x+99))=f(x)+C, t...

If `int(dx)/((x+100)sqrt(x+99))=f(x)+C`, then `f(x)` is

A

`2(x+100)^(1//2)`

B

`3(x+100)^(1//2)`

C

`2tan^(-1)(sqrt(x+99))`

D

`2tan^(-1)(sqrt(x+100))`

Text Solution

Verified by Experts

The correct Answer is:
C
Let `l=int(dx)/([(sqrt(x+99))^(2)+1]sqrt(x+99))`
Put`" "sqrt(x+99)= t rArr (1)/(sqrt(x+99))dx=2dt`
`therefore" "=int(2dt)/(t^(2)+1)=2 tan^(-1)t+C`
`=2tan^(-1)sqrt((x+99))+C`
But`" "2tan^(-1)sqrt(x+99)+C=f(x)+C" [given]"`
`therefore" "f(x)=2 tan^(-1)(sqrt(x+99))`
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