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int("sin"(5x)/(3))/("sin"(x)/(2))dx is e...

`int("sin"(5x)/(3))/("sin"(x)/(2))dx` is equal to (where, C is a constant of integration)

A

`(1)/(sqrt(2))(x+log_(e)|cos(x-(pi)/(4))|)+c`

B

`(1)/(sqrt(2))(x-log_(e)|sin(x-(pi)/(4))|)+c`

C

`sqrt(2)(x+log_(e)|sin(x-(pi)/(4))|)+c`

D

`(1)/(sqrt(2))(x+log_(e)|sin(x-(pi)/(4))|)+c`

Text Solution

Verified by Experts

The correct Answer is:
D
` int(sinx)/(sin(x-pi//4))dx=int(sin((x-(pi)/(4))+(pi)/(4)))/(sin(x-(pi)/(4)))dx`
`=(1)/(sqrt(2))int(sin(x-(pi)/(4))+cos(x-(pi)/(4)))/(sin(x-(pi)/(4)))dx`
`=(1)/(sqrt(2))int(1+cot sin(x-(pi)/(4)))dx`
`=(1)/(sqrt(2))(x+log_(e)|sin(x-(pi)/(4))|)+c`
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