Use the substitutions in Equations (1)–(4) to evaluate the int...

Question

Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.
∫ cos t dt / (1 - cos t)

∫ cos t dt / (1 - cos t)

Accepted Answer

Welcome back everyone. Compute the integral cosinet divided by 1 plus cosine Tdt. For this problem, let's go ahead and rewrite the integral cosine c divided by. One plus cosine cdc. What we can do is simply add and subtract 1 in the numerator so that we're going to get 1 plus cosine t minus 1. Divided by 1 plus cosine CDT, which essentially allows us to apply the properties of fractions we can divide selectively, right? So this is going to be equal to integral of 1 minus 1 divided by 1 plus cosinetdt. And now we can split the integral into two parts. We're integrating dt, or basically 1dt minus the integral of 1 divided by 1 plus cosine t dt. So let's focus on the second integral because the first one is simply t plus a constant of integration. To integrate dt divided by 1 plus cosinec, we could use trigonometric identities. Specifically, we can say that 1 plus cosinet can be written as 2 cosines squared off t divided by 2 using the half angle formula, right? So now the integral of. The sea. Divided by 2 cosine squared of t divided by 2. Can also be written as one half integral of 2 and squared. Of t divided by 2 dt, right? Because 1 divided by cosine squared is 2 squared. And now this is an elementary integral, we can apply the substitution. Let's suppose that you is. T divided by 2. Then we can show that the derivative of u with respect to t is the derivative of 1/2 t, which is simply 1/2. And therefore we can solve for DC. DC multiplied by 1 is equal to 2 multiplied by DU. So now the integral becomes 1/2. Integral of c square of u multiplied by 2 duu, we can just write dU and we can factor out 2. And 2 divided by 2 is equal to 1, so we're basically integrating sequin square of uDU. And this is an elementary integral. Its value is tangent of you. Plus a constant of integration. Let's call it C1 for now. And replacing you with t divided by 2, we get tangent oft divided by 2 plus a constant of integration C1. Now let's combine the results and identify the value of the overall integral i. So i is going to be equal to the integral of the t, which is just t. Minus Tangent oft divided by 2 plus a constant of integration c. We have two constants of integration from each indefinite integral, and we can just combine those into the overall constant c. And that's our final answer. Thank you for watching.

Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.∫ cos t dt / (1 - cos t)

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Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.
∫ cos t dt / (1 - cos t)