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Evaluate int_(0)^(int) (tan^(-1)(ax))/(xsqrt(1-x^(2)))dx, 'a'being parameter. |
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Answer» Solution :Let `I(a) = underset(0)overset(1)int(tan^(-1)(AX))/(xsqrt(1-x^(2)))dx rArr (dI(a))/(DA) = underset(0)overset(1)(int)(x)/((1+a^(2)x^(2)))(1)/(xsqrt(1 x^(2)))dx=underset(0)overset(1)int(dx)/((1+a^(2)x^(2))sqrt(1-x^(2)))` Put `x = SINT rArr dx = cos t dt` L.L : `x = 0 rArr t = 0` U.L. `x = 1 rArr t = (PI)/(2)` `(dI(a))/(da) = underset(0)overset(pi/2)(int)(sec^(2)tdt)/(1+(1+a^(2))tan^(2)t)= (1)/(sqrt(1+a^(2))) tan^(-1)(sqrt(1+a^(2))TANT)]_(0)^(pi/2) = (1)/(sqrt(1+a^(2))).pi/2` `rArr I(a) = pi/2ln (a+sqrt(1+a^(2)))+c` But `I(0)= 0 rArr c = 0 rArr I(a) = (pi)/(2) ln (a+sqrt(1+a^(2)))` |
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