InterviewSolution
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InterviewSolution
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Evaluate `int(dx)/((1+5sin^(2)x))`. |
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Answer» On dividing num . And denom by `cos^(2)x` , we get `int(dx)/((1+5sin^(2)x))=int((1//cos^(2)x))/(((1)/(cos^(2)x)+5*(sin^(2)x)/(cos^(2)x)))dx` `=int(sec^(2)x)/((sec^(2)x5tan^(2)x))dx=int(sec^(2)x)/({(1+tan^(2)x)+5tan^(2)x})dx` `=int(sec^(2)x)/((1+6tan^(2)x))dx=int(dt)/((1+6t^(2)))`, where tan =t `=(1)/(6)int(dt)/(((1)/(6)+t^(2)))=(1)/(6)*int(dt)/({((1)/(sqrt(6)))^(2)+t^(2)})` `=(1)/(6)*(1)/((1//sqrt(6)))"tan"^(-1)(t)/((1//sqrt(6)))+C=(1)/(sqrt(6))tan^(-1)(sqrt(6t))+C` `=(1)/(sqrt(6))tan^(-1)(sqrt(6)tanx)+C`. |
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