C. Hinsley

14 September 2020

Note: In Surely You're Joking, Mr. Feynman!, Feynman mentions the two books from which he learned calculus. The former is "Calculus for the Practical Man" by J.E. Thompson and the latter (the one from which he learned Liebniz' integral rule) is "Advanced Calculus" by F.S. Woods. Both of these texts are intended to prepare the reader for the use of calculus in an applied setting, and I think each of these have a distinct clarity missing from modern texts on the subject. It may be worthwhile to read these texts, even for one who has already studied calculus. The basis for technique in particular can be found on page 141 of the 1934 edition of Woods' text.

These exercises lend themselves to solution by Feynman's technique, which you can see worked out for example problems at various other sources. I found that these sources only had a handful of problems each, so I decided to compile all I could find in one location.

$$ (1).\int_0^1 \frac{t^3-1}{\ln t}\ \text{d}t $$

$$ (2). \int_0^1 (x\ln x)^{50}\ \text{d}x $$

$$ (3). \int_0^\infty e^{-\frac{x^2}{2}}\ \text{d}x $$

$$ (4).\int_0^{2\pi} e^{\cos \theta}\cos(\sin\theta)\ \text{d}\theta $$

$$ (5). \int_0^\infty \frac{\sin x}{x}\ \text{d}x $$

$$ (6). \int_0^{\frac{\pi}{2}} \frac{x}{\tan x}\ \text{d}x $$

$$ (7). \int_0^\infty \frac{\ln(1+x^2)}{1 + x^2}\ \text{d}x $$

$$ (8). \int_0^1 \frac{x-1}{\ln x}\ \text{d}x $$

$$ (9). \int_0^1 \frac{\ln(x+1)}{x^2+1}\ \text{d}x $$