Finding Integrals Using Trigonometric SubstitutionsDate: 10/2/95 at 8:12:18 From: roberto manzueta Subject: What is the integral of 1/(x(1-x))^(1/2)? Dr. Math, I'm currently taking Calculus II. I have not been able to integrate the above function with the methods I have learned so far. The answer to this problem is the arcsin [something] because I saw it at the back of my text (I do not remember that "something" right now). Anyway, how can I integrate this function using only the derivative of arcsin or arctan? Date: 10/20/95 at 14:11:18 From: Doctor Ken Subject: Re: What is the integral of 1/(x(1-x))^(1/2)? Hello! Here's a hint about how to do your integral. Notice that the denominator of the fraction looks like the square root of a quadratic polynomial. This should make the buzzers in your brain go off and scream "TRIG SUBSTITUTION!!" Of course, it isn't in _exactly_ the right form for a trig substitution. Remember that the derivative of Arcsin(t) is 1/Sqrt{1 - t^2). Well, your problem is how to turn the inside part, x(1-x) into something of the form 1-t^2. Hint: write x(1-x) as 1 - (1 - x(1-x)) and then complete the square on the 1 - x(1-x) part! Hope this is enough for you to go on. Good luck, and write back if you need more help! -Doctor Ken, The Geometry Forum |
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