Jim Coroneos' 100 Integrals ~ 031 ~ ∫sinx/(5 + 3cosx).dx

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Partly to honour Jim, and partly to fulfil an international need, I have decided to produce 100 videos, showing how to solve his 100 integration 'problems.' I hope you find the videos useful!

This thirty-first problem is to evaluate ∫sinx/(5 + 3cosx).dx

This integral looks similar to the previous two, but we find that Jim has been having a joke with us! In the middle of a long series of quite difficult integrals, he presents one that looks deceptively difficult. In fact, it is quite elementary, and can be solved in two steps!

Once you have gained experience, when you encounter the integral of a fraction, one of the first quick tests that you will apply is to see whether the numerator is the derivative of the denominator. This is the pattern that gives rise to logarithmic functions. It so happens that the derivative of (5 + 3cosx) is -3sinx so, by multiplying the numerator of our integral by -3, we have the perfect structure for a logarithmic answer! We compensate by multiplying the integral by -1/3 and the result follows in just one step.

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Thank you.
  1. Crystal Clear Maths
  2. Integral (Literature Subject)
  3. Jim Coroneos
  4. James Coroneos
  5. Mathematics (Field Of Study)
  6. math
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Комментарии
Автор

God bless you Professor, you are the most helpful online math professor. Thank you, and best wishes for you.

propagandadrunken
Автор

Funny anecdote, dear Graeme. Even more funny integral. Only, I would say that strictly pedagogical way of showing how to solve this integral is to use "t substitution" with  5 + 3cosx = t. But, with pracitce comes the ability to recognize the form and structure and the ability to do such substitution in the head. Thank you for some humorous side of math.

MrVoayer
Автор

Thanks for the reply, sir. What part of Australia are you in? I visited Townsville once when I was in the Marines. I love everything about Australia, well except maybe the vegemite lol!

propagandadrunken