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Evaluate: `intsqrt(secx-1) dx` |
Answer» ` I=int sqrt(secx-1)dx=int sqrt((1-cosx)/(cosx))dx` `=int sqrt(((1-cosx))/(cosx)xx((1+cosx))/((1+cosx)))dx` `=int sqrt((1-cos^(2)x)/(cosx+cos^(2)x))dx` `=int(sinx)/(sqrt(cos^(2)x+cosx))dx` Let `cosx=t.` Then `d(cosx)=dt` or `-sinx dx=dt.` Therefore, `I=int(-dt)/(sqrt(t^(2)+t))` `=-int(dt)/(sqrt((t+(1)/(2))^(2)-((1)/(2))^(2)))` `=-log|(t+(1)/(2))+sqrt((t+(1)/(2))^(2)-((1)/(2))^(2))|+C` ` =-log|(t+(1)/(2))+sqrt(t^(2)+t)|+C` `=-log|(cosx +(1)/(2))+sqrt(cos^(2)x+cosx)|+C` |
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