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int_(pi/2)^(pi) e^(x) ((1-sinx)/(1-cosx)) dx |
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Answer» Solution :Let `I= int_(pi//2)^(pi) E^(x) ((1-sin x)/(1-cos x))DX` `=int_(pi//2)^(pi)e^(x)[(1-2SIN((x)/(2))cos ((x)/(2)))/(2sin^(2)((x)/(2)))]dx` `=int_(pi//2)^(pi) e^(x) ((1)/(2)"cosec"^(2) (x)/(2) -cot .(x)/(2))dx` `int_(pi//2)^(pi) e^(x) (-cot .(x)/(2) +(1)/(2) " cosex"^(2) .(x)/(2))dx` `[underset(inte^(x) {f(x) +f(x)}dx =e^(x)f(x))("Here " (d)/(dx) (-cot.(x)/(2))=(1)/(2)"cosec"^(2).(x)/(2))]` `:. I=[e^(x) (-cot .(x)/(2))]_(pi//2)^(pi)` `=-e^(x) cot ((pi)/(2))-[-e^(pi//2)cot ((pi)/(4))]` `=-e^(pi).0+e^(pi//2).1=e^(pi//2)` |
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