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in Definite Integrals by (100k points)
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`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

1 Answer

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Best answer
Correct Answer - 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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