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int secx/(sqrt(sin(2x+alpha)+sinalpha))d...

`int secx/(sqrt(sin(2x+alpha)+sinalpha))dx`

A

`sqrt(secalpha(tanx+tanalpha))+C`

B

`sqrt(2 secalpha(tanx-tanalpha))+C`

C

`sqrt(2 secalpha(tanalpha-tanalpha))+C`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
a
Let `I=int(secx)/(sqrt(2x+alpha+sinalpha))dx`
`rArr I=int(secx)/(sqrt(2 sin(x+alpha)cosx))dx`
`rArrI=(1)/(sqrt(2))int(sec^(2)x)/(sqrt(tanx cos alpha+sinalpha))dx`
`rArrI=(1)/(sqrt(2)cosalpha)int(1)/(sqrt(tanx cosalpha+sinalpha))d(tanx cosalpha+sinalpha)`
`rArrI=(1)/(sqrt(2)cos alpha)xx2sqrt(tan x cosalpha+sinalpha)+C`
`rArr I=sqrt(2sec^(2)alpha(tan xcosalpha+sinalpha))+C`
`rArrI=sqrt(2secalpha(tanx+tanalpha))+C`.
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