# Tagged Questions

For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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### About odd functions and improper integrals e.g. $\int^{\infty}_{-\infty}\sin x \; dx$

Does $\displaystyle \int^{\infty}_{-\infty}\sin x \; dx$ converge? Since $\sin x$ is an odd function, and we know that in definite integrals $\displaystyle \int^{a}_{-a}\sin x \; dx=0$ then does ...
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### Proof $\{(x,y,z)|4x^2+9y^2+16z^2<1\}$ is an open set

In order to prove that the points $(x,y,z)$ such that $$4x^2+9y^2+16z^2<1$$ form an open set, I tried this: Pick a generic point of the ellipsoid, lets say $$4x^2+9y^2+16z^2$$ Now, I'll form ...
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### What is the ratio of the intensities of the two sounds?

1. Suppose that a jet engine at 50 meters has a decibel level of 130, and a normal conversation at 1 meter has a decibel level of 60. What is the ratio of the intensities of the two sounds? we ...
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### Find $\int_a^b \sin |x| \, \mathrm{d}x$

How to find the integral $$\int_a^b \sin |x| \, \mathrm{d}x \,?$$ I'm able to obtain definite integral of form $\int_a^b \lvert\sin x \rvert \, \mathrm{d}x$ but not when the modulus operator is ...
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### Why doesn't L'Hopital's rule work in this case?

I have a very simple question. Suppose I want to evaluate this limit: $$\lim_{x\to \infty} \frac{x}{x-\sin x}$$ It is easy to evaluate this limit using the Squeeze theorem (the answer is $1$). But ...
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### Help in understanding a limit involving an integral.

$$I_n = \int^1_0\frac{x^n}{ax+b}dx$$ Where: $n \in N$ ; $a,b \in (0,\infty)$" Find $\Xi$, where: $$\Xi=\lim_{n \to \infty}nI_n$$
### Evaluate $\int_{0}^{\frac{\pi}{4}}\frac{\sec^2 \theta }{(1-\tan \theta )}\ d \theta$
Evaluate $$\int_{0}^{\frac{\pi}{4}}\frac{\sec^2 \theta }{(1-\tan \theta )}\ d \theta$$ Here's my attempt: $$u=1-tan \theta \implies -du=\sec^2 \theta d \theta$$ Substituting back in, I get this: ...