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The following integral int(pi//4)^(pi//2...

The following integral `int_(pi//4)^(pi//2)(2cosec x)^(17) dx dx` is equal to

A

`int_(0)^("log"(1+sqrt(2)))2(e^(u)+e^(-u))^(16)` du

B

`int_(0)^("log"(1+sqrt(2)))2(e^(u)+e^(-u))^(17)` du

C

`int_(0)^("log"(1+sqrt(2)))2(e^(u)-e^(-u))^(17)` du

D

`int_(0)^("log"(1+sqrt(2)))2(e^(u)-e^(-u))^(16)` du

Text Solution

Verified by Experts

The correct Answer is:
A
PLAN This type of question can be done using appropriate substitution.
Given , `I=int_(pi//4)^(pi//2)(2 " cosec " x)^(17)dx`
`=int_(pi//4)^(pi//2)(2^(17)("cosec " x)^(16)("cosec " x+ cot x))/(("cosec " x+cotx))`
Let `"cosec " x+cotx =t`
`rArr(-"cosec "x*cotx-"cosec"^(2)x)dx = dt`
and `"cosec " x-cotx=1//t`
`rArr2 " cosec "x=t+(1)/(t)`
` :.I=-int_(sqrt(2)+1)^(1)2^(17)((t+(1)/(t))/(2))^(16)(dt)/(t)`
Let `t=e^(u)rArrdt=e^(u)du`.
When `t=1,e^(u)=1rArru=0`
and when `t=sqrt(2)+1,e^(u)=sqrt(2)+1`
`rArru=In(sqrt(2)+1)`
`rArrI=-int_("In" (sqrt(2)+1))^(0)2(e^(u)+e^(-u))^(16)(e^(u)du)/(e^(u))`
`=2int_(0)^("in"(sqrt2+1))(e^(u)+e^(-u))^(16)du`
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