What Derivative Rule does Trig Substitution match up with?

Question

I was thinking about the Integral Rules and trying to understand why they work. It seems to me like all Integral Rules should have a Derivative Rule counterpart:

• U Substitution: Chain Rule. The chain rule says: $\frac d{dx}f(g(x)) = f'(g(x))\times g'(x)$ and U Sub removes a function and its derivative from the integral:$\int g'(x)f'(g(x))dx = \int f'(u)du$
• Integration by Parts: Product/Quotient Rule. Example: $\int x \arctan{x}dx$. Thinking using product rule, it can be seen that $\frac d{dx}\left(\frac12x^2 \arctan x\right)$ $=$ $\displaystyle x \arctan x + \frac x{2(1+x^2)}$. So $\int x \arctan x dx = \int \left(x \arctan x + \frac x{2(1+x^2)}\right)dx - \int \frac x{2(1+x^2)}dx = \frac12x^2\arctan x - \int \frac x{2(1+x^2)}dx$ which is the rule for Integration by Parts.
• Partial Fractions: Not really any derivative rule here; this is basically manipulating the function with algebra to make it easier to integrate.
• Trig Substitution: ?

What Derivative rule matches up with Trig Substitution? Why does Trig Substitution work?

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It's easy to see how the first two methods were invented. How was Trig Substitution invented?

Answer 1

I think it's very fruitful to think of rules of integration as corresponding to rules of derivatives. But I suspect that trig substitutions don't fit this pattern. A couple of simple ones might - for example, instead of memorizing the derivatives of arctan and arcsin, you can do trig substitutions in those integrals and "discover" their antiderivatives. But in general, I see trig substitution as a way of making integration more robust in the stage of development after we got as much mileage as we can out of reversing differentiation rules. We noticed that we have techniques for integrating lots of rational functions; we tried to think of ways that we could improve our toolkit to be able to integrate all rational functions. Trig substitutions (and the method of partial fractions) were what we came up with.

Answer 2

Trig substitution has nothing to do with a specific derivative rule, other than the chain rule. It's just a smart u-sub. But how did trig sub even come about? Trig sub is a method of substitution that allows us to make use of trigonometric identities, often the Pythagorean Identities:

$$ \sin^2 + \cos^2 = 1\\ 1 + \cot^2 = \csc^2\\ \tan^2 + 1 = \sec^2 $$

This can often make for simpler integrals (especially ones with ugly square roots), such as if we have a $\sqrt{a^2-x^2}$, we can substitute (among many possible substitutions) $x = a \cos \theta$, reducing the sqrt down to $a\sin\theta$. Basically, trig sub is a method of removing square roots. It might also be a method to do other things as well.