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if int0^k (dx)/(2+8x^2)=pi/16 then find ...

if `int_0^k (dx)/(2+8x^2)=pi/16` then find the value of `k`

Text Solution

Verified by Experts

Let `I=int_(0)^(h)(1)/(2+8x^(2))dx`
`=(1)/(8)int_(0)^(h)(1)/(x^(2)+((1)/(2))^(2))dx`
`=(1)/(8)xx(1)/(((1)/(2)))[tan^(-1)((x)/(1//2))]_(0)^(h)`
`=(1)/(4)[tan^(-1)2x]_(0)^(h)`
`=(1)/(4)[tan^(-1)2h-tan^(-1)0]=(1)/(4)tan^(-1)2h`
`therefore" "I=(pi)/(16)" gives "(1)/(4)tan^(-1)2h=(pi)/(16)`
`therefore" "tan^(-1)2h=(pi)/(4)" "therefore 2h=tan.(pi)/(4)=1`
`therefore h=(1)/(2)`.
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