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Evaluate: ("lim")(nvecoo)[1/(n^2)sec^2 1...

Evaluate: `("lim")_(nvecoo)[1/(n^2)sec^2 1/(n^2)+2//n^2sec^2 4/(n^2)++1/nsec^2 1]`

A

`1/2 tan1`

B

tan 1

C

`1/2 "cosec"1`

D

`1/2 sec 1`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `A=underset(nrarroo)(lim)(1)/(n)`
`[(1)/(n)sec^(2)((1)/(n))^(2)+(2)/(n)sec^(2)((2)/(n))^(2)+...+(n)/(n)sec^(2)((n)/(n))^(2)]`
`=underset(nrarroo)(lim)(1)/(n)sum_(r=1)^(n)((r)/(n))sec^(2)((r)/(n))^(2)`
`rArr" "A=int_(0)^(1)x sec^(2)(x)^(2)dx`
`"Put "x^(2)=t" "rArr" "xdx=(dt)/(2)`
`therefore" "A=(1)/(2)int_(0)^(1)sec^(2)tdt=(1)/(2)[tant]_(0)^(1)=(1)/(2)tan1`
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-DEFINITE INTEGRALS-PRACTICE EXERCISE (Exercise 2) (MISCELLANEOUS PROBLEMS)
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  6. If f(x) = f(a+x) and int(0)^(a) f(x) dx = k, "then" int (0)^(na) f(x) ...

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  11. If int (0)^(pi//2) sin^(6) dx = (5pi)/32 , then the value of int (...

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  12. int (0)^(pi//4) [ sqrt(tan x)+ sqrt(cot x)] dx is equal to

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  13. Evaluate the following limit: lim(nto oo)[(n!)/(n^(n))]^(1//n)

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  14. Let a, b and c be non - zero real numbers such that int (0)^(3) (3ax...

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  15. The value of integral sum (k=1)^(n) int (0)^(1) f(k - 1+x) dx is

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  16. Ifint(sinx)^1t^2f(t)dt=1=1-s inx ,w h e r ex in (0,pi/2), then find t...

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