I posted this question on the Mathematics Stack Exchange a while ago, and got no responses, so I thought I would ask it here. I'm looking for any real-life applications of integration by substitution where the integrand is unknown as a way of motivating the topic, as opposed to giving repetitive questions which can be answered by typing the integral into WolframAlpha.
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1$\begingroup$ You need a lot of substitution when you learn line integrals, but this is more advanced and not so good for motivation. $\endgroup$– András BátkaiJun 12, 2019 at 9:07
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1$\begingroup$ I don't think that you should be so wary of Wolfram Alpha danger, when looking for motivational exercises. By nature, they may be simple or even iconic. That's a feature, not a bug. (Even pre-comps, you could make the same stipulation bit with tables versus comps. But you have to realize that for new students, it's NEW in a different way from you.) Remember what is new, interesting to YOU is different than what a neophyte needs. $\endgroup$– guestJun 12, 2019 at 22:02
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2 Answers
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This is probably too abstract for students just learning about integration by substitution. But convolution is a "real-world application of integration," finding applications in such things as image processing. One defines
$$(f*g)(t)=\int_{-\infty}^{\infty} f(\tau)g(t-\tau)\, d\tau$$
and then to show $f*g=g*f$ you use integration by substitution.
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You could look at separable differential equations:
Given a differential equation of the form $$g(y)y'(x) = f(x),$$ integrating between $x_0$ and x on both side and integration by substitution with $u = y(x)$ gives
$$\int_{x_0}^xg(y)y'(x) dx = \int_{x_0}^xf(x)dx $$
$$\int_{y(x_0)}^{y(x)}g(u) du = \int_{x_0}^xf(x)dx,$$
which allows to solve the differential equation, provided we can find antiderivatives $G$ and $F$ and invert $G$. It is probably possible to find a physical context where a separable equation appears if you want to insist on the "real-life" aspect.
