# Browsing category calculus

## A “creative” integral

An interesting “creative” integral pointed my way by the marvellous @DavidKButlerUoA: Find $\int {\frac{1}{(1+x^2)^2}} \dx$ There are “proper” ways to do this - in his tweet, David shows a clever way to do it by parts, and suggests a trig substitution as an alternative. I want to show a third

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When faced with something like $\int \frac{1}{\sqrt{1+x^2}} \dx$, my first instinct has usually been to panic, and then to try trig (or hyperbolic) substitutions more or less at random. But is there a better way? There are six such integrals altogether: $\int \frac{1}{\sqrt{1-x^2}} \dx = \arcsin(x) + C$ $\int \frac{-1}{\sqrt{1-x^2}} Read More ## ILATE vs LIATE Some time ago, I recommended the mnemonic "LIATE" for integration by parts. Since you have a choice of which thing to integrate and which to differentiate, it makes little sense to pick something that's hard to integrate as the thing to integrate. With that in mind, you would look down Read More ## Ask Uncle Colin: a nasty integral Dear Uncle Colin, How would you integrate$e^x \sin(x)$(with respect to$x$, obviously)? - Difficult Integral, Just Kan't See The Right Answer Hi, DIJKSTRA, and thanks for your message! As seems to be the way recently, there are several ways to approach this. My favourite way One of the Read More ## Square wheels on a round(ish) floor The ever-challenging Adam Atkinson, having noticed my attention to the "impossible" New Zealand exams, pointed me at a tricky question from an Italian exam which asked students to verify that, to give a smooth ride on a bike with square wheels (of side length 2), the height of the floor Read More ## Ask Uncle Colin: A multi-cubic integral Dear Uncle Colin, I need to calculate$\int x^3 (x^3+1) (x^3 + 2)^{\frac 13} \dx$and it's giving me a headache! Can you help? I've Blundered Using Parts, Rolled Out Fourier Expansions... Nothing! Hi, IBUPROFEN, and thanks for your message! That’s a bit of a brute, but it can be Read More ## The Sneakiest Integral I’ve Ever Done Once upon a time1, @dragon_dodo asked me to help with:$\int_{- \piby 2}^{\piby 2} \frac{1}{2007^x+1} \frac{\sin^{2008}(x)}{\sin^{2008}(x)+\cos^{2008}(x)} \dx$. Heaven - or the other place - only knows where she got that thing from. Where do you even begin? My usual approach when I don't know where to start is to start Read More ## Ask Uncle Colin: Evil in integral form Dear Uncle Colin, I've been set an evil integral:$\int_0^\piby{4} \frac{\sqrt{3}}{2 + \sin(2x)}\d x$. There is a hint to use the substitution$\tan(x) = \frac{1}{2}\left( -1 + \sqrt{3}\tan(\theta)\right)$, but I can't see how that helps in the slightest. -- Let's Integrate Everything! Hello, LIE, and thank you for your message! Read More ## Revisiting Basel Some while ago, I showed a slightly dicey proof of the Basel Problem identity,$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac {\pi^2}{6}$, and invited readers to share other proofs with me. My old friend Jean Reinaud stepped up to the mark with an exercise from his undergraduate textbook: The French isn't that difficult, Read More ## An Integral Diversion The brilliant @dragon_dodo sent me this puzzle: Evaluate$\int_0^1 \left(1-x^\frac{1}{7}\right)^3 - \left(1-x^\frac{1}{3}\right)^7 \d x\$. I'm not going to give you the solution right now; that will come after I've rambled for a bit. After I'd solved the puzzle (see below), I wondered what each of the integrals actually evaluated to.

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