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int(0)^(x) (dx)/(1-2a cos x+a^(2)) is eq...

`int_(0)^(x) (dx)/(1-2a cos x+a^(2))` is equal to

A

`(pi)/(2(1-a^(2)))`

B

`pi(1-a^(2))`

C

`(pi)/(1-a^(2))`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
C
Let ` l = int _(0)^(pi) (dx)/(1-2a cos x+a^(2))`
` int _(0)^(pi) (dx)/((1+a^(2)) (cos^(2).x/2 +sin^(2). x/2 ) - 2a (cos^(2). x/2 - sin^(2) .x/2 ) )`
` = int _(0)^(pi) (dx)/((1-a^(2))cos^(2) . x/2 + (1+ a^(2)) sin^(2). x/2 ) `
` = 2/ ((1+a^(2)) )int _(0)^(oo)(dt)/({((1-a))/((1+a))}^(2)+t^(2))`
` [ t = tan .x/2 rArr dt = 1/2 sec^(2) . x/2 dx] `
` l = 2/(1+a)^(2).((1+a))/((1-a)) [ tan^(-1) ((1+a)/(1-a).t ) ] _(0)^(oo) `
` l = 2/((1-a^(2))) [ tan ^(-1) oo - tan ^(-1) 0 ] = pi/(1-a^(2))`
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MHTCET PREVIOUS YEAR PAPERS AND PRACTICE PAPERS-DEFINITE INTEGRALS-MHT CET Corner
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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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  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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