Browsing category calculus

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

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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

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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

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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

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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

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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!

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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,

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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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Ask Uncle Colin: A Polar Expression

Dear Uncle Colin, I was asked to find the tangent to the curve $r=\frac{8}{\theta}$ at the point where $\theta = \frac{\pi}{2}$. I worked out $\dydx = \frac{ \frac{8 \left(\theta \cos(\theta)-\sin(\theta)\right)}{\theta^2}}{\frac{-8\left(\theta \sin(\theta)+\cos(\theta)\right)}{\theta^2} }$, which simplifies to $ -\frac{\theta \cos(\theta)-\sin(\theta)} {\theta \sin(\theta)-\cos(\theta)}$. Evaluated at $\theta = \frac{\pi}{2}$, that gives $\dydx=\frac{2}{\pi}$ and a

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A curve-sketching masterclass

An implicit differentiation question dealt with $y^4 – 2x^2 + 8xy^2 + 9 = 0$. Differentiating it is easy enough for a competent A-level student – but what does the curve look like? That requires a bit more thought. My usual approach to sketching a function uses a structure I

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