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Evaluate `(int_(0)^(pi)/2)(tanxdx)/(1+m^(2)tan^(2)x)` |
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Answer» Let `I=int_(0)^(pi//2)(tan xdx)/(1+m^(2)tan^(2)x)` `=int_(0)^(pi//2)((sinx)/(cosx))/(1+m^(2) . (sin^(2) x)/(cos^(2)x)) dx` `= int_(0)^(pi//2)(sinx cos x dx)/(1-sin^(2)x+m^(2)sin^(2)x) dx` `=int_(0)^(pi//2)(sinx cosx)/(1+(m^(2)-1)sin^(2)x)` Put `sin^(2)x=t` `implies2sin x cos x dx=dt` when `xto0`, then `t to 0` and `xto pi/2,` then `t to 1` `:. I=1/2 int_(0)^(1)(dt)/(1+(m^(2)-1)t)` `=1/2[1/(m^(2)-1)log(1+(m^(2)-1)t)]_(0)^(1)` `=1/(2(m^(2)-1))[log(1+(m^(2)-1))-log1]` `=(log m^(2))/(2(m^(2)-1))` `=(2log|m|)/(2(m^(2)-1))` `=(log|m|)/(m^(2)-1)` |
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