Practice Questions for test

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March 23rd 2008, 01:48 PM
polymerase

Practice Questions for test

Hey Everyone,

First let me start by saying, I'm not sure if such a book exist.

Does anyone know of any good textbooks/books/reference books for first year (single variable, early transcendentals) Calculus, that have more challenging/not common/bit more thinking and application questions? Or tough definate/indefinate integrals.

For example questions like, $\int ^b_a \frac{dx}{\sqrt{(x-a)(b-x)}}$ or If $\int ^{\frac{\pi}{4}}_0 \tan^6 x\sec x\;dx=L$ find the value of $\int ^{\frac{\pi}{4}}_0 \tan^8 x\sec x\;dx$ in terms of $L$ .

Thanks!
March 23rd 2008, 02:49 PM
Krizalid

Prove for $m,n\ge0$ that $<br /> \int_0^1 {x^m (1 - x)^n \,dx} = \frac{{m!n!}}<br /> {{(m + n + 1)!}}.$ This is actually a direct application of Beta function.

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This is also a really nice problem, the first time I saw it, I attempted to compute both values of the integrals.

No matter what $a$ is. Let $I=\int_0^1\frac{e^u}{1+u}\,du$ & $J=\int_{a-1}^a\frac{e^{-x}}{x-a-1}\,dx.$ Assume that $\lambda\cdot I=J,$ compute $\lambda.$

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I recently posted (in another forum) a proof for $\left(\frac12\right)!=\frac{\sqrt\pi}2,$ that also could be another nice problem to do. (You can see its solution here. It's written in spanish, but I think you can get the main ideas.)

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Compute $\int_0^1\ln x\ln(1-x)\,dx.$ When I saw this problem, I developed a solution which involves double integration & series application. (If someone is interested, see my solution here.)

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Evaluate $\lim_{n \to \infty } \left\{ {\frac{1}{{n^2 }}\sum\limits_{k = 0}^{n - 1} {\left[ {k\int_k^{k + 1} {\sqrt {(x - k)(k + 1 - x)} \,dx} } \right]} } \right\}.$ This one I love it! It's easy and nice!

I have many others problems, but I don't remember them now, but try these ones.
March 23rd 2008, 03:18 PM
polymerase

Thank You VERY much; I appreciate the effort and time! ....However...
these are not exactly the types of questions I'm looking for. I'm looking for easier questions to do with integration and series. So I guess Krizalid maybe instead of your interesting questions give me YOUR "very easy" question.
March 23rd 2008, 03:25 PM
Krizalid

Let me tell ya something: you asked for tough integrals and I haven't given you tough integrals.

Take a look at my problems, last two involves series (first one more than the second one), and the penultimate problem not necessarily has to be computed with double integration, I actually made that solution to simplify things. The second one is a really nice problem, if you don't try it, you can't say it is a "hard" problem. As for the third problem, it's also a nice problem which involves integration, take a look at my solution, if you want to, I can translate my message.

I'd like to see more "easy-interesting" problems from another members.

P.S.: by the way, it's "definite integral." :)