Browsing category integration

Ask Uncle Colin: A trigonometric integral

Dear Uncle Colin, I'm trying to find a definite integral: $\int_0^\pi \sin(kx) \sin(mx) \dx$, where $m$ and $k$ are positive integers and the answer needs to be simplified as far as possible. I've wound up with $\left[\frac{ (k+m) \sin((k-m) \pi) - (k-m)\sin((k+m)\pi) }{2(k-m)(k+m)}\right]$, but it's been marked wrong. -- Flat

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Ask Uncle Colin: Integration rules

Dear Uncle Colin, Why can't I work out $\int \left( \ln(x) \right)^2 \dx$ using the reverse chain rule? -- Previously Acceptable, Reasonable Technique Stumbles Hello, PARTS, There are two answers to this: the first is, you can't use the reverse chain rule -- which I learned as 'function-derivative' when I

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Where do the suvat equations come from?

Most of the suvat equations are pretty easy to derive, as soon as you realise acceleration ($a$, assumed constant) is the derivative of velocity ($v$) with respect to time, and velocity is the derivative of position ($s$), also with respect to time. For example: $ a = \diff{v}{t}$ $ \int_0^t

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Ask Uncle Colin: an integral that’s giving me a headache

Dear Uncle Colin, I've been trying to work out $I = \int_0^{\frac \pi 4} x \frac{\sin(x)}{\cos^3(x)} \d x$ for hours. It's the fifth time this week I've been up until the small hours working on integration and it's affecting my work and home life. I'm worried I'm becoming a calcoholic.

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There’s More Than One Way To Do It: Differential Equations

Until fairly recently, I had always done the kind of differential equations you see in Core 4 the same way: separate, integrate, substitute, celebrate. I have taught any number of students the dance; many of them have boogie-woogied their way into a correct answer in exams. But there's a variation

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Ask Uncle Colin: A Sticky Integral

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can. Dear Uncle Colin, I've been trying to integrate $\int \frac{x^2}{x-1}

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Integrating $\sec^4(x)$

A student asks: How do you integrate $\int_{0}^{\frac{\pi}{4}} \sec^4(x) \d x$? Yuk. Let me say that again for good measure: yuk. That's going to need a trigonometric identity and, I think, a substitution. But that's ok: we can do that. Let's roll up our sleeves. Step 1: get rid of

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“A little biter of a question”

This problem came via the lovely @realityminus3 and caused me no end of problems - although I got there in the end. I thought it'd be useful to look at not just the answer, but the mistakes I made on the way. Maths is usually presented as 'here's what you

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That pesky constant

A student asks: I've got to work out: $\int \cosec^2(x) \cot(x) \d x$. I did it letting $u = \cosec(x)$ and got an answer -- but when I did it with $u = \cot(x)$, I got something else. What gives? Ah! A substitution question! My favourite -- and it sounds

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A student asks: Why is there a $+c$ when you integrate?

A student asks: We've just started integration and I don't understand why there's always a $+c$ - I understand it's a constant, I just don't understand why it's there! Great question! The simple answer is, because constants vanish when you differentiate, they have to appear when you integrate - it's

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