Question

If I = `int 1/(2p) sqrt((p - 1)/(p + 1))dp` = f(p) + c, then f(p) is equal to ______.

Options

• `1/2 ℓn[p - sqrt(p^2 - 1)]`

• `1/2 cos^-1 p + 1/2 sec^-1p`

• `ℓn sqrt(p + sqrt(p^2 - 1)) - 1/2 sec^-1p`

• none of the above

Answer 1

If I = `int 1/(2p) sqrt((p - 1)/(p + 1)) dp` = f(p) + c, then f(p) is equal to `underlinebb(ℓn sqrt(p + sqrt(p^2 - 1)) - 1/2 sec^-1p)`.

Explanation:

I = `int 1/(2p) sqrt((p - 1)/(p + 1)) dp` 

= `1/2int (p - 1)/(psqrt((p + 1)(p - 1)) dp`

= `1/2int (pdp)/(psqrt(p^2 - 1)) - 1/2 int (dp)/(psqrt(p^2 - 1))`

= `1/2int (dp)/sqrt(p^2 - 1) - 1/2int (dp)/(psqrt(p^2 - 1)`

= `1/2 log_e (p + sqrt(p^2 - 1)) - 1/2 sec^-1p`

`\implies` f(p) = `logsqrt(p + sqrt(p^2 - 1)) - 1/2 sec^-1p`