What is the integral of the function (cos2x) ^3/2. (cosx) where the limits are taken ...?

We are given the definite integral [math]I = \displaystyle \int_0^{\pi/4} \cos^{\frac{3}{2}}(2x) \cos{x} \, dx. \tag*{}[/math] Using the double angle identity for cosine and then make the subs…

Top stories for Long   What is the integral of the function (cos2x) ^3/2. (cosx) where the limits are taken from 0 to pi/4? Brian Sittinger, PhD Mathematics, University of California, Santa Barbara (2006) • Answered August 15 We are given the definite integral [math]I = \displaystyle \int_0^{\pi/4} \cos^{\frac{3}{2}}(2x) \cos{x} \, dx. \tag*{}[/math] Using the double angle identity for cosine and then make the substitution [math]t = \sqrt{2} \sin{x}[/math], we obtain [math]\begin{align*} I &= \displaystyle \int_0^{\pi/4} (1 - 2 \sin^2{x})^{\frac{3}{2}} \cos{x} \, dx\\ &= \int_0^1 (1 - t^2)^{\frac{3}{2}} \cdot \frac{1}{\sqrt{2}} \, dt\\ &= \frac{1}{\sqrt{2}} \int_0^1 (1 - t^2)^{\frac{3}{2}} \, dt. \end{align*} \tag*{}[/math] Next, we let [math]w = t^2[/math] to convert the integral into a s... Read more » 13   What part of the US is the worst that you won't even want to visit again? Lee Kamps, Benefit Specialist at Independent Agent • Answered June 12 I have visited 46 out of the fifty states. I define visiting a state as spending a night there, driving across or through the state or visiting a major attraction or city i... Read more » 114 35   Whether you like Donald Trump or not, is he right about global warming being a hoax and a scare tactic by the left because oceans will rise only an eighth of an inch in about 355 years? Kevin Peterlinz, Research Scientist (1983-present) • Answered February 8 As a physical chemist, I have been following the peer reviewed scientific papers since the early 1980's - and the data and calculations in detail since 1987 Among others rea... Read more » 835 232   Why do employers have the right to reject applications from candidates who recently graduated from universities in the US? These students spent years of their lives to get a competitive degree and now some companies still reject them. Pamela Bruce, Reluctant educator on the political and economic scene. • Answered July 23 Well, Vanessa, if I may be blunt. You say you have a masters in engineering. However, when I read your questions, they read like you have less maturity than my almost teena... Read more » 501 33   Is having children worth it? Jens Adler Nielsen, PhD Applied Mathematics & Physics, Danish Technical University • Answered August 12 I saw a study comparing the happiness of people with and without children. If I remember correctly, the study was done in Holland, Denmark, and Australia. The study reached... Read more » 191 35   What is something men will never understand? Rose, B.A. Psychology & Political Science, Western Washington University (2008) • Updated Fri This…. If you are above 40 and you are trying to date a woman your age… You men really need to understand one fundamental thing… It is highly unlikely that you're going to fi... Read more » 1.6K 377   Why can't we produce our own electricity when we only receive half of 1% from Canada? Frank Lucrezi, studied read history science at at reads alot • Answered March 26 We were promised that and lower rates in PA a number of years ago. The GOP pushed through our congress deregulating electricity. The electric companies had to sell their po... Read more » 776 77   How inaccurate are the dinosaurs from Jurassic park? Charcoal Jerky • Answered March 11 At that time? Pretty darn accurate based on what we know. I mean this is how they thought the T-rex looked back in the 70's: Walking upright and dragging it's tail behind it.... Read more » 3.1K 132   Was Lyndon Johnson happy when he realized that John Kennedy was dead and he was going to be President? Will Scathlocke • Answered September 29, 2017 A chance, off-hand remark of LBJ's may speak volumes about what he really felt about JFK's assassination. When he was president, LBJ struck up a warm friendship with Harry ... Read more » 4.2K 367   What is the best thing to say when a police officer asks you to step out of your vehicle? Thomas Brand, studied at Mississippi State University • Answered August 7 Say "I am going to reach down to my right hip and release the seatbelt. Then I will have to reach to the door to open it. Please stop intentionally standing in the way of t... Read more » 117 11   If gambling is a game of probability, why aren't mathematicians the richest people in the world? Asked in Ask Daddy • Answered by Dormas • Wed This person is Chris "Jesus" Ferguson. Ferguson, whose parents are both mathematicians, holds a doctorate in information engineering from the University of California, Los A... Read more » 2.5K 88   Why are most people broke? Rhonda Lester • Answered June 8, 2017 I work as a financial advisor for a Fortune 500 company, and I have clients from a broad range of income brackets. Part of my job involves evaluating income and spending/sa... Read more » 550 50 Read more in your feed

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