Indefinite IntegralsDate: 07/04/2002 at 14:51:38 From: Anirban Bhattacharyya Subject: Indefinite Integral 1. Integral dx/(sin x)^5 + (cos x)^5. 2. Integral dx/(sin x)^3 + (cos x)^5. Date: 07/10/2002 at 10:52:02 From: Doctor Nitrogen Subject: Re: Indefinite Integral Hello, Anirban: The two integrals you submitted: 1. Integral dx/((sinx)^5 + (cosx^5)), 2. Integral dx/((sinx^3) + (cosx)^5), can be solved by using various trigonometric identities and by then employing certain substitutions. Let's start with (1). First, notice that (sinx)^5 + (cosx)^5 = (cosx)^5(1 + (tanx)^5), using the identity (sinx)^5/(cosx)^5 = (tanx)^5. Here all you have done is factored out (cosx)^5, which leaves 1 + (sinx)^5/(cosx)^5 in the parentheses. So now the integral becomes Integral dx/(cosx)^5(1 + (tanx)^5). Using the fact that 1/(cosx)^5 = (secx)^5, change the integral to: Integral (secx)^5dx/(1 + (tanx)^5). Here is where you can try a substitution. Let U = tanx dU = (secx)^2dx cosx = 1/sqrt(1 + U^2) secx = sqrt(1 + U^2). This changes the integral to: Integral (secx)^5dx/(1 + (tanx)^5) = Integral (sqrt(1 + U^2))^3dU/(1 + U^5) since (secx)^5dx = (secx)^3*(secx)^2dx and dU = (secx)^2dx. Now notice that (sqrt(1 + U^2))^3 = (1 + U^2)^(3/2.) Unfortunately, you are not done yet. You must use two more substitutions. For the first substitution try V = 1 + U^2 U = sqrt(V - 1) dV = 2UdU dU = 1/2(dV/U) = 1/2(dV/sqrt(V - 1)) This changes the integral to: (1/2) * Integral (V^3/2)dV/(sqrt(V - 1))^6 = (1/2) * Integral(V^3/2)dV/(V - 1)^3. The second substitution is: V^1/2 = W V = W^2 dV = 2WdW, so that you finally come up with the integral: Integral W^4dW/(W + 1)^3(W - 1)^3. From this point on, you must use partial fractions to get your answer. You can solve (2) by a similar method, only it leads to a different numerator: From (sinx)^3 + (cosx)^5, factor out cosx^5 in the denominator. For the denominator you will get (cosx)^5(1 + (sinx)^3/(cosx)^5) = (cosx)^5(1 + (tanx)^3*(secx)^2). Use the identity (tanx)^2 = (secx)^2 - 1 to change the integral to: Integral (secx^5)dx/(1 + (tanx)^3*(secx)^2). Now use the substitution U = tanx dU = secx^2dx secx = sqrt(1 + U^2), and further substitutions similar to those in the first solution to get your answer. You will again have to use partial fractions after you make the last substitutions. And don't forget to "backtrack" through all the substitutions you made when you get the integral value in other 'dummy' variables, so you will wind up again with all expressions in x. I hope this was helpful. If you have any more questions, please contact Dr. Math again. - Doctor Nitrogen, The Math Forum http://mathforum.org/dr.math/ |
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