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Evaluate the following : `int_(0)^(pi//2)(dx)/(a^(2)cos^(2)x+b^(2)sin^(2)x)` |
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Answer» Correct Answer - `(pi)/(2ab)` `I=int_(0)^(pi//2) (dx)/(a^(2)cos^(2)x+b^(2)sin^(2)x)` `=int_(0)^(pi//2) (sec^(2)x dx)/(a^(2)+b^(2)tan^(2)x)=1/(b^(2)) int_(0)^(pi//2) (sec^(2) xdx)/((a/b)^(2)+tan^(2)x)` Put `tan x=z` or `sec^(2) x dx=dz` When `x=0, z=0, xto (pi)/2, zto oo` `:.I=1/(b^(2)) int_(0)^(oo) (dz)/((a/b)^(2)+z^(2))` `=1/(b^(2))1/(a/b)|"tan"^(-1)z/(a//b)|_(0)^(oo)` `=1/(ab)[tan^(-1)oo-tan^(-1)0]` `=1/(ab) (pi)/2` `=(pi)/(2ab)` |
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