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The value of int (0)^(pi//2) (2^(sinx))/...

The value of `int _(0)^(pi//2) (2^(sinx))/(2^(sinx)+2^(cosx))dx` is

A

2

B

`pi`

C

`pi/4`

D

`2pi`

Text Solution

Verified by Experts

The correct Answer is:
C
Let `l=int_(0)^(pi//2)(2^(sinx))/(2^(sinx)+2^(cosx))dx" ...(i)"`
`rArr" "l=int_(0)^(pi//2)(2^(cosx))/(2^(cosx)+2^(sinx))dx" ...(ii)"`
On adding Eqs. (i) and (ii), we get
`2l=int_(0)^(pi//2)1dx=(pi)/(2)`
`rArr" l=(pi)/(4)`
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-DEFINITE INTEGRALS-PRACTICE EXERCISE (Exercise 2) (MISCELLANEOUS PROBLEMS)
  1. The value of the integral int3^6 sqrtx/(sqrt(9-x)+sqrtx)dx is

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  2. int-1^1 (17x^5-x^4+29x^3-31x+1)/(x^2+1)dx is equal to

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  3. The value of int (0)^(pi//2) (2^(sinx))/(2^(sinx)+2^(cosx))dx is

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  4. Evaluate int0^(pi/4) log(1+tanx)dx

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  5. The value of the integral int (a)^(b) (sqrt(x)dx)/(sqrt(x)+sqrt(1+b-x...

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  6. int (0)^(pi//2) (cos x - sin x)/(1+cos x sin x)dx is equal to

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  7. Evaluate: ("lim")(nvecoo)[1/(n^2)sec^2 1/(n^2)+2//n^2sec^2 4/(n^2)++1/...

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  8. The value of the integral int (0)^(pi//2) log | tan x| dx is

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  9. int (-pi//2)^(pi//2) sin^(4) x cos^(6) x dx is equal to

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  10. 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. The value of int(0)^(sqrt(2)) [ x^(2)] dx where [*] is the greatest i...

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  12. int0^pi(x dx)/(a^2cos^2x+b^2sin^2x)

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  13. l=int(-2)^(1)(tan^(-1)x+"cot"^(-1)(1)/(x))dx is equal to

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  14. int(0)^(pi) x sin^(4) x dx is equal to

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

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

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

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

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

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  20. 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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