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int(pi/3)^(pi/2) sqrt(1+cosx)/(1-cosx)^(...

`int_(pi/3)^(pi/2) sqrt(1+cosx)/(1-cosx)^(5/2) dx`

A

`(3)/(64)`

B

`-(3)/(64)`

C

`(-3)/(64)`

D

`-(1)/(128)`

Text Solution

Verified by Experts

The correct Answer is:
B
` int _(pi//3) ^(pi//2) (sqrt(1+cos x))/((1-cos x)^(5//2) ) dx `
` int _(pi//3)^(pi//2) (sqrt(2 )cos. x/2)/(2^(5//2) sin^(5). x/2) dx = 1/4 int _(pi//3)^(pi//2) (cos.x/2)/(sin^(5).x/2)dx`
Put `sin. x/2 = t`
` rArr = 1/2 cos. x/2 dx= dt`
Also at ` x= pi/3 , t = sin. pi/6 = 1/2 `
and at ` x = pi/2 , t = sin. pi/4 = 1/(sqrt(2))`
So integral ` = 1/2 int _(1//2) ^(1//sqrt(2))(dt)/(t^(5) ) = 1/2 [ (-1)/(4t^(4))]_(1//2)^(-1//sqrt(2))`
` = (-1)/8 [ 1/4 - 1/16 ] = (-3)/128`
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-DEFINITE INTEGRALS-MHT CET Corner
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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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  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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  21. int (1)^(2)e^(x) (1/x - 1/(x^(2)))dx is qual to

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