グローバルビジネスで役立つ数学（19）三角関数の定積分（英語版）※Thanks for three likes for my note.

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
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