Hi, there!
It was not until the first grade of junior high school that I met my best friend
for the first time. Pardon me to hide his exact name to protect his individual
information; his name was “HT.”
The reason why I talked about him today is that it was he that had taught me
the bliss of studying. I was almost a drop-out student, but he talked me into
learning in a library after school and helped me with homework. Without his
thoughtfulness, I would never have started to post on my math blog.
In return for his kindness in the past, I want to post the word problem related
to definite integrals of trigonometric functions. “‘HT, this is me, who used to
walk home from school together while reciting Indian greetings like a sutra.
I’ve become a bit more studious than before, you see.”
Here comes the word problem.
Evaluate the following integral.
\(\int_{0}^{\frac{π}{4}}\frac{16cos(2t)}{\sqrt{18 – 10sin(2t)}}dt\)
First of all, we need to do the substitution.
\(u = 18 – 10sin(2t)\)
Here is the actual substitution for this problem.
\(du = -20cos(2t)dt = -\frac{1}{20}du\)
\(t = 0: u = 18\), then \(t = \frac{π}{4}: u = 8\)
Let’s convert the limits to \(u\)’s to avoid having to deal with the back substi-
tution after doing the integral. Here is the integral after substitution.
\(\int_{0}^{\frac{π}{4}}\frac{16cos(2t)}{\sqrt{18 – 10sin(2t)}}dt\)
\(= -\frac{16}{20}\int_{18}^{8}u^{-\frac{1}{2}}du\)
The integral is then,
\(\int_{0}^{\frac{π}{4}}\frac{16cos(2t)}{\sqrt{18 – 10sin(2t)}}dt\)
\(= -\frac{4}{5}\int_{18}^{8}u^{-\frac{1}{2}}du\)
\(= [-\frac{8}{5}u^{\frac{1}{2}}]_{18}^{8}\)
\(= [-\frac{8}{5}\sqrt{8} – (-\frac{8}{5}\sqrt{18})]_{18}^{8}\)
\(= \frac{24\sqrt{2} – 16\sqrt{2}}{5}\)
\(= \frac{8\sqrt{2}}{5}\)
“HT, please contact me if you found my miscalculations.
I might solicit you to teach me again in the library.”
Stay tuned, and expect to see my next post.
Keep well.
Frank Yoshida
￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣
【グローバルビジネスで役立つ数学】でもっと学習する b＾＾)
【参考図書】『もう一度高校数学』（著者：高橋一雄氏）株式会社日本実業出版社
【レッスン】私のオンライン英語レッスンをご希望の方はこちらをご覧ください。
【コンテント】当サイトで提供する情報はその正確性と最新性の確保に努めていま
すが完全さを保証するものではありません。当サイトの内容に関するいかなる間
違いについても一切の責任を負うものではありません。
￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣
只今、人気ブログランキングに参加しています。
今日の［実践数学の達人］のランキングは――
