InterviewSolution
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InterviewSolution
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Evaluate the following integral: `int_0^(pi//4)(sqrt(t a n x)+sqrt(cotx))dx` |
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Answer» We have `I=int_(0)^(pi//2){(sqrt(sinx))/(sqrt(cosx))+(sqrt(cosx))/(sqrt(sinx))}dx=int_(0)^(pi//2)((sinx+cosx))/(sqrt(sinx+cosx))dx` `=sqrt(2)*int_(0)^(pi//2)((sinx+cosx))/sqrt(2sinxcosx)dx=sqrt(2)*int_(0)^(pi//2)((sinx+cosx))/(sqrt(1-(sinx-cosx)^(2)))dx` Put `(sinx-cosx)=t` and `(cosx+sinx)dx=dt`. Also, `[x=0impliest=-1]` and `[x=(pi)/(2)impliest=1]`. `:.I=sqrt(2)*int_(-1)^(1)(dt)/(sqrt(1-t^(2)))=sqrt(2)[sin^(-1)t]_(-1)^(1)` `=sqrt(2){sin^(-1)(1)-sin^(-1)(-1)}=sqrt(2){2sin^(-1)(1)}` `=(sqrt(2)xx2xx(pi)/(2))=sqrt(2)pi`. |
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