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int (0)^(pi//4) log ((sin x+ cos x)/(cos...

` int _(0)^(pi//4) log ((sin x+ cos x)/(cosx))dx ` is equal to

A

`(pi)/(8)log2`

B

`(pi)/(4)log2`

C

`log2`

D

`(pi)/(2)log2`

Text Solution

Verified by Experts

The correct Answer is:
A
Let ` l = int _(0)^(pi//4) log ((sin x + cos x)/(cos x)) dx`
` l =- int _(0)^(pi//4) log (1+tan x ) dx" " `
` = int _(0)^(pi//4) log [ 1 + tan (pi/4 -x) ] dx`
` = int _(0)^(pi//4) log (1+ (tan pi//4-tan x)/(1+tan pi //4 tanx))dx`
` = int _(0)^(pi//4) log (1+ (1-tanx)/(1+tan x)) dx`
` rArr l = int _(0)^(pi//4) log (2/(1+tan x))dx" "` ...(ii)
On adding Eqs. (i) and (ii) , we get
` 2l = int _(0)^(pi//4) log 2 dx = log 2 (pi/4)`
` rArr l = pi/8 log 2 `
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-DEFINITE INTEGRALS-MHT CET Corner
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  2. int (-pi/2)^(pi/2)log((2-sin x)/(2+sinx))dx is equal to

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  3. int (0)^(pi //2)((root(n)(secx))/(root(n)(secx)+root(n)("cosec"x)))dx=

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  4. The value of int 0 ^ 1 x ^ 2 ( 1 - x ^ 2 ) ^ (3//2 ) dx ...

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  5. The value of int0^oox/((1+x)(x^2+1))dx is

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  6. Evaluate int(0)^(pi)(x dx)/(1+cos alpha sin x),where 0lt alpha lt pi.

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  7. int(pi//2)^(pi//2)(cosx)/(1+e^(x))dx is equal to

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  8. int(0)^(pi//2)(1)/((1+tanx))dx=?

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  9. If int(0)^(1) tan^(-1) x dx = p , then the value of int(0)^(1) tan^(-1...

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  10. The value of int (0)^(pi//2) log ("cosec "x) dx is

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  11. Which of the following is true ?

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  12. int(0)^(5) 1/((x-1)(x-2))dx is equal to

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  13. int(pi/4)^(pi/2) e^x(logsinx+cotx)dx

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  14. The value of int(0)^(pi) x sin^(3) x dx is

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  15. The value of int0 ^(pi/2) (cos3x+1)/(cosx - 1) dx is equal to

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  16. The value of underset(0)overset(1)int tan^(-1) ((2x-1)/(1+x-x^(2)))dx ...

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  17. If f is a continous function, then

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  18. The value of int(-pi)^(pi) sin^(3) x cos^(2) x dx is equal to

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  19. The value of int(-1)^(1) log ((x-1)/(x+1))dx is

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  20. int(pi//6)^(pi//3)(1)/((1+sqrt(tanx)))dx=(pi)/(12)

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  21. int (1)^(2)e^(x) (1/x - 1/(x^(2)))dx is qual to

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