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If int(0)^(k)(dx)/(2+18x^(2))=(pi)/(24),...

If `int_(0)^(k)(dx)/(2+18x^(2))=(pi)/(24)`, then the value of k is

A

3

B

4

C

`(1)/(3)`

D

`(1)/(4)`

Text Solution

Verified by Experts

The correct Answer is:
C
We have,
`int_(0)^(k)(dx)/(2+18x^(2))=(pi)/(24)`
`implies(1)/(18)int_(0)^(k)(dx)/(((1)/(3))^(2)+x^(2))=(pi)/(24)`
`implies(1)/(18)xx(1)/((1)/(3))[tan^(-1)3x]_(0)^(k)=(pi)/(24)`
`implies[tan^(-1)3k-10^(-1)0]=(pi)/(4)`
`impliestan^(-1)3k=(pi)/(4)`
`implies3k="tan"(pi)/(4)`
`implies3k=1impliesk=(1)/(3)`
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Knowledge Check

  • If int_(0) ^(k) (dx)/(2+18 x^(2))=(pi)/(24), then the value of k is

    A
    `3`
    B
    `4`
    C
    `(1)/(3)`
    D
    `(1)/(4)`

  • If int_(0)^(a)sqrt((a-x)/(x))dx=(K)/(2), then K=. . .

    A
    `(pia)/(2)`
    B
    `(5pia)/(2)`
    C
    `(3pia)/(2)`
    D
    `pia`

  • If int_(0)^(pi)((x)/(1+sinx))^(2) dx=A, then the value for int_(0)^(pi)(2x^(2). cos^(2)x//2)/((1+ sin x^(2)))dx is equal to

    A
    `A+2pi-pi^(2)`
    B
    `A-2pi+pi^(2)`
    C
    `2pi-A-pi^(2)`
    D
    None of these

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