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If a, b and c are real numbers, then the...

If a, b and c are real numbers, then the value of `lim_(trarr0) ln((1)/(t)int_(0)^(t)(1+asinbx)^(c//x)dx)` equals

A

abc

B

`(ab)/(c)`

C

`(bc)/(a)`

D

`(ca)/(b)`

Text Solution

Verified by Experts

The correct Answer is:
A
`L=underset(trarr0)(lim)log_(e)((1)/(t)int_(0)^(t)(1+a sinbx)^(c//x)dx)((0)/(0)"form")`
`=loe_(e)(underset(trarr0)(lim)(int_(0)^(t)(1+a sin bx)^(c//x)dx)/(t))" (Using L' Hopital Rule)"`
`=loe_(e)(underset(trarr0)(lim)((1+a sin bt)^(c//x)/(1))`
`=log_(e)[(underset(trarr0)(lim)(1+asin bt)^((1)/(a sin bt)))^((ac sin bt)/(1))]`
`=log_(e)e^(abc)`
`=abc`
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