# Tagged Questions

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### Proving the $u$-substitution formula for Lebesgue integrals

Following the proof at Wikipedia, I'm trying to verify the proof of the substitution rule for integrals under the (fairly simple) assumptions that $\phi:[a,b]\to(c,d)$ is continuous on $[a,b]$ and ...
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### Projection measures and integrability

Let $(M, \mathcal{A}, \mu)$ a probability space, $Y$ compact metric space. Consider $\mathcal{M}(\mu)$ be the space of probability measures $\eta$ on $M\times Y$ such that $\pi_{*}\eta=\mu$ where ...
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### Unique solution for $\int_x^1 f(t) dt = 2x$ and $|x| < \epsilon$

Let $f$ be continuous on $\mathbb{R}$ such that $$f(0) \neq -2 \quad\text{ and } \quad \int_0^1 f(t) = 0.$$ Show that there exists $\epsilon > 0$ such that the equation $$\int_x^1 f(t) dt = 2x$$ ...
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### Integration by change the variable

Let, $\int_{-1}^1\sqrt{1+e^x}\operatorname{dx}$. Write as an integral of a rational function and compute it. Suggest: change the variable in order to eliminate the square root. My work was: ...
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### If $f(x)$ is integrable on $[a,b]$ then $c\cdot f(x)$ is also integrable and $\int_a^b c\cdot f(x) dx=c\cdot \int_a^b f(x) dx$

I proved the first part of this theorem which says that $c\cdot f(x)$ is integrable,but how to prove that $\int_a^b c\cdot f(x) dx=c\cdot \int_a^b f(x) dx$? Maybe it provides a bit help if i tell how ...
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### Proof verification: $\int_a^x f(t) \text{dt}=0$, $f$ is continuous at $x$. Prove that $f(x)=0$

Let $f:[a,b]\to R$ be an integrable function such that for all $x \in[a,b]$, we have $\int_a^x f(t) \text{dt}=0$. Show that if $f$ is continuous at $x \in [a,b]$, then $f(x)=0$. My attempt: argue ...
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### Would like to compute the limit of some integral

I was working on a exercise where the goal was to compute the following limit, $$\lim_{n\rightarrow\infty}\int_{\mathbb R}e^{-|x|n}e^{-\frac{x^2}{2}}dx$$ and some tutor of mine claimed that the limit ...
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### Area of solid of known cross-section

I was looking at surface areas of solids of known cross-section (that is, you take a region in the xy plane, set up cross sections perpendicular to the region that are defined as a function of x, and ...
We are given vector space of polynomials over $\mathbb R$ of two variables with powers not higher than 2013. Let's consider subspace $V$ which contains such polynomials $f$, so following holds for ...