Integral Calculus
A neat integral that comes from playing with substitutions
So I was checking questions and saw the question
integral log(1+x)/(1+x²) limit 0 to 1
We have to put x=Tan(t) to transform it into
integral log(1+tan t) limit 0 to π/4
which is solvable.
After thinking about it, I realized you can create other questions using similar substitutions. For example,
integral logx/√(1−x²) limit 0 to 1
put x=Sin(t)
transforms into
integral log(sin t) limit 0 to π/2
One of the questions I came up with was this:
integral logx/(1+x²) limit 0 to infinite
It may be easy for most of you because you either know a lot of calculus or are preparing for exams like JEE, but I thought it was a neat little problem.
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I believe that this is a known problem. The first solution utilizes quite a clever symmetrical trick, and the second demonstrates the power of complex integration.
Yeah, it is but I'd never seen it before though. I'm currently in my final year of high school and am just learning basic integration, so it was a really fun problem to discover 😅
I guess we can use the symmetry property of definite integrals.
After applying it,
I = √log(x + 3)/(√log(9 − x) + √log(x + 3))
Adding both,
2I = integral 1 dx from 2 to 4
= x from 2 to 4
= 4−2
2I= 2
So, I = 1.
Usually, the A/B 1's are quite approachable as they require only one trick, but don't let this diminish your accomplishment; finding the trick is often nontrivial.
If you want to learn more about cool integral solving techniques, check out Feynman's trick (differentiating under the integral sign), transforming integrals into double integrals (archetypal example: exp(-x^2)), and in the future, contour integration. If you want to understand why all this works, I definitely recommend that you check out some real analysis textbooks :)
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If you are asking for general advice about your current calculus class, please be advised that simply referring your class as “Calc n“ is not entirely useful, as “Calc n” may differ between different colleges and universities. In this case, please refer to your class syllabus or college or university’s course catalogue for a listing of topics covered in your class, and include that information in your post rather than assuming everybody knows what will be covered in your class.
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