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

Evaluate `int_0^(pi/4) log(1+tanx)dx`

A

`pi/8 log_(e)2`

B

`pi/4 log_(2)e`

C

`pi/4log_(e)2`

D

`pi/8 log_(e)(1/2)`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `l=int_(0)^(pi//4)log(1+tanx)dx" ...(i)"`
`rArr" "l=int_(0)^(pi//4)log[1+tan((pi)/(4)x)]dx`
`" "=int_(0)^(pi//4)log[1+(1-tanx)/(1+tanx)]dx`
`" "=int_(0)^(pi//4)log2dx-int_(0)^(pi//4)log(1+tanx)dx`
`rArr" "l=log2[x]_(0)^(pi//4)-l`
`rArr" "2l=(pi)/(4)log_(e)2`
`rArr" "l=(pi)/(8)log_(e)2`
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-DEFINITE INTEGRALS-PRACTICE EXERCISE (Exercise 2) (MISCELLANEOUS PROBLEMS)
  1. int-1^1 (17x^5-x^4+29x^3-31x+1)/(x^2+1)dx is equal to

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

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

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

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

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

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

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

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  9. If f(x) = f(a+x) and int(0)^(a) f(x) dx = k, "then" int (0)^(na) f(x) ...

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

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

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

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

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

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

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

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

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

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  19. 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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  20. The value of int (2)^(4) { |x-2|+|x-3|} dx is

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