A Trigonometry Integral Requiring Two SubstitutionsDate: 05/03/98 at 17:14:16 From: Paul. Subject: Calculus Integration. What is the integral of sqrt(1 + sin(x)), where sqrt stands for "square root of"? -Paul Date: 05/04/98 at 16:58:23 From: Doctor Sam Subject: Re: Calculus Integration. Paul, This is a tricky problem. It will take (I think) two different substitutions. We want to find: INT sqrt(1 + sin x) dx I am going to try substituting u = sin(x) to try to remove the trig function. When you make a substitution, you must also substitute for dx. So: u = sin(x) and du = cos(x) dx This gives dx = du/cos(x), and changes the integral to sqrt(1 + u) INT ----------- du cos(x) This is no good. We need to get an integral in terms of the u variable alone. Here's where a little right-triangle trigonometry can help. We made the substitution u = sin(x), so we can visualize a triangle with an acute angle x whose sine is u. Here is one such triangle: /| / | / | / | 1 / | u / | /x | -------- Now we can use the Pythagorean Theorem to find the third side, and then the cosine of x. The third side is sqrt(1 - u^2), and so: cos(x) = sqrt(1 - u^2) Our integral is now: sqrt(1 + u) INT ------------- du sqrt(1 - u^2) Now I can't help but notice that 1 - u^2 = (1 - u)(1 + u) so this fraction simplifies to: 1 INT ----------- du sqrt(1 - u) We are almost done. We have now transformed our trig integral into an algebraic integral. Now a second substitution: w = 1 - u should finish the job. If w = 1 - u, then dw = -du, so du = -dw. This gives: 1 - INT ------- dw sqrt(w) Interpret this as w^(-1/2), and we can use the formula for antidifferentiating u^n: 1 - INT ------ dw = -2w^(1/2) + C sqrt(w) Now change back from w to u using w = 1 - u: -2w^(1/2) = -2 sqrt(1 - u) + C And now change back from u to x using u = sin(x): -2w^(1/2) = -2 sqrt(1 - u) + C = -2 sqrt(1 - sin(x)) + C I hope that helps. Doctor Sam, The Math Forum http://mathforum.org/dr.math/ Date: 05/08/98 at 00:26:54 From: Paul Oommen Subject: Re: Calculus Integration. Thanks for the answer. -Paul |
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