49–63. {Use of Tech} Integrating with a CAS Use a computer alg...

Question

49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.
52. ∫ from 0 to π/2 of cos⁶x dx

52. ∫ from 0 to π/2 of cos⁶x dx

Accepted Answer

Welcome back, everyone. In this problem, we want to find the exact value of the following definite integral the integral of cost to the 4th of X with respect to X between 0 and a half of pi. A says it's 3/16 of pi, B 55 pi divided by 32, C3/8 of pi, and D, 5 pi divided by 64. Now to help us find this exact value, it would be a good idea for us to use the poor reduction formula on our call center. Recall that by the power reduction formula. OK. Then We can say that the cosine to the 4th of X. Is going to be equal to 3/8 plus 1/2 + 2 X plus 1/8 of 4 X. So if we apply that to our integral, OK, that means our integral of cost to the 4th of X, let me write my limits properly, of cost to the 4th of X with respect to X is going to be equal to the integral between the same limits of 3/8 plus 1/2 of 2 X sorry, plus 1/8 of 4 X. All with respect to X. And now when we use what we know about integration, we that means we can apply the integral to each term in our sum. In other words, that means we'll be finding, uh, once we put out our constants, 3/8 of the integral with respect to X, OK. Plus a half of the integral. Of course 2 X with respect to X. Plus 1/8 of the integral. Of costs for X with respect to X. So to help us figure that out, let us go ahead and find the definite integral of each term and then combine the results. So integrating each term, OK. Integrating each term. Then, for starters, we'll be finding 3/8 of the integral between our limits or the definite integral with respect to X. And now we know that's gonna be 3/8 of X between our limits. And then we substitute to our limits. Then that's gonna be 3/8 of a half of pi minus 3/8 of 0. 3/8 of a half of pi is 3/16 of pi and 3/8 of 0 is 0. So for the first term, the definite integral is 3/16 of pi. Let's move on to our next two terms. Now for a half of the integral. Of cost 2 X with respect to X, OK, then we're going to end up with a 1/2 multiplied by a half of sin 2 X. Between 0 and 1/2 of pie. A half multiplied by 1/2 is a quarter. I know when we apply our limits in our integral, that's gonna be the sign of 2 multiplied by 1/2 of pi, which is just going to be pi minus the sign of 2 multiplied by 0, which is going to be 0. No, the sign of pi is 0 and the sign of 0 is 0, and a quarter of 0 minus 0 equals 0. No 4 or 3rd term, that is 1/8 of the integral. Of costs for X with respect to X. Now, here, we're gonna have 18, OK? And we know that it's going to be multiplying a quarter of sine 4 X between our limits. And we know that 1/8 multiplied by a quarters, 1 divided by 32, and know for our terms by evaluating between a definite integral, that is going to be the sign of 4 multiplied by a half of pi or 2 pi, OK, minus the sign of 4 multiplied by 0, which is 0. We know that the sign of 2 pi is 0, the sign of 0 is 0, and at 1 divided by 32 multiplied by 0 equals 0. So now that we have our 3 terms. From the definite integral, if we combine our results, then that means we end up with 3 16th of pi. Plus 0 + 0, which is equal to 3/16 of pi. Therefore, the exact value of the integral of cost to the fourth of X with respect to x between 0 and 2 of pi is equal to 3/16 of pi. If we look back at our answer choices, that means A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.
52. ∫ from 0 to π/2 of cos⁶x dx

Key Concepts

Definite Integral

Computer Algebra System (CAS)

Trigonometric Integrals

Watch next

49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.52. ∫ from 0 to π/2 of cos⁶x dx

Key Concepts

Definite Integral

Computer Algebra System (CAS)

Trigonometric Integrals

Watch next

49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.
52. ∫ from 0 to π/2 of cos⁶x dx