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Calculus II - Trigonometric Substitutions, Explanation Please

  1. Aug 2, 2011 #1
    Evaluate

    integral sqrt(9-x^2)/x dx

    I get to here

    -3 ln|csc(sin^-1(x/2)+cot(sin^-1(x/3))|+3cos(sin^-1(x/3)+c

    I plugged in for theta because originally I had this

    -3 ln|csc(theta)+cot(theta)|+3cos(theta)+c

    and I set x equal to a expression

    x=3sin(theta)

    Hence i had to change my equation back to terms of x because the original integral was in terms of so by solving for theta

    theta=sin^-1(x/3)

    -3 ln|csc(theta)+cot(theta)|+3cos(theta)+c = -3 ln|csc(sin^-1(x/2)+cot(sin^-1(x/3))|+3cos(sin^-1(x/3)+c

    but I'm lost as to why

    -3 ln|csc(sin^-1(x/2)+cot(sin^-1(x/3))|+3cos(sin^-1(x/3)+c=-3ln((sqrt(9-x^2)+3)/x)+sqrt(9-x^2)+c

    and was hoping someone could explain why
     
  2. jcsd
  3. Aug 2, 2011 #2
    nevermind i think i got it it's a triangle
     
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