# Tagged Questions

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### How to prove that $\max\{f,g\}$ is Riemann integrable? [duplicate]

If f(x) and g(x) are Riemann integrable in [a,b], why $h(x)=\max\{f(x),g(x)\}$ is still Riemann integrable in [a,b]? Or maybe it is wrong?
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If f(x) and g(x) are integrable in [a,b], can we say that f(x)g(x) is still integrable in [a,b]? I am referring to Riemann integration!
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### Finding all continuous solutions to an integral

I need help with both parts of this problem. Part (i) seems obvious, because the integrand $f(t)$ would become $F(t)$, which is obviously differentiable because its derivative is $f(t)$ by ...
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### Solving integral $\int \frac{x+\sqrt{1+x+x^2}}{1+x+\sqrt{1+x+x^2}}\:\mathrm{d}x$

there is integral $$\int \frac{x+\sqrt{1+x+x^2}}{1+x+\sqrt{1+x+x^2}}\:\mathrm{d}x$$ i am trying to separate this : $$=\int \mathrm{d}x -\int \frac{\mathrm{d}x}{1+x+\sqrt{1+x+x^2}}$$ but have no idea ...
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### Prove with Lebesgue’s Criterion for integrablility that the composition $f\circ g$ is integrable

I have this homework question regarding Lebesgue's criterion for integrability and could use a bit of help. I'm not sure if my proof is entirely correct or formal enough. Here is said question: ...
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### Let $S_n:= \frac{b-a}{n}\sum_{i=1}^{n}f(t_{i,n})$. Prove: $\lim_{n\to\infty}S_n = \int_a^bf(x)\ dx$.

I will post the assignment and then my attempt at solving it. Let $a,b \in \mathbb{R}$ with $a<b$ and let $f: [a,b] \rightarrow \mathbb{R}$ be a continous function. We'll now define a sequence ...
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### integral of sin(x) to the power 2014

For a course in Complex Analysis we're tasked to find the integral of \begin{align*} \int_0^{2 \pi} (\sin\theta)^{2014} d \theta \end{align*} but I'm a bit stumped so far on how to do this. What I've ...
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### Surface Area cut from cone by cylinder

I genuinely have no idea how to do this. I've been struggling on this problem for about two hours now (more hours last night), and am not getting anywhere. The question is: Find the surface area ...
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### Evaluating improper integral

Im trying to evaluate the improper integral $$\int_{0}^{\infty}\left( \frac{e^{i \omega t}+e^{-i \omega t}}{2}\right) e^{-st} dt$$, where $\omega$ and $s$ are real positive constants and ...
Show the area enclosed by $r=a(p+qcos\theta)=\dfrac{2p^2+q^2}{2}\pi a^2$. Due to the form of the equation, the curve is either a cardioid or an egg shape - either way, the boundaries of the integral ...