Questions tagged [substitution]
Questions that involve a replacement of variable(s) in an expression or a formula.
2,080 questions
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Integration by substitution with $x$ in the denominator
I'm going to calculate the following indefinite integral
$$\int 3x\sqrt{5x^2+7}dx$$
using the change of variable $u=5x^2+7$, $du=10xdx$. From this last expression, can I solve for $dx$, that is, $\...
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Question about the axiom of substitution in Tao’s Analysis I
I'm quoting a sentence from page 57 of Analysis I by Terence Tao:
“We observe that functions obey the axiom of substitution:
if $x = x'$, then $f(x) = f(x')$ (why?).”
My question is: is it possible ...
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Determine if $\sum_{n = - \infty} ^ {\infty} e^{-(2t -n)}u(2t -n)$ is periodic. If so, what is its period?
I am given a continuous function $x(t) = \sum_{n = - \infty} ^ {\infty} e^{-(2t -n)}u(2t -n)$, where $u(t)$ is the unit step function, and asked to find its period, if it exists. I'm not sure how to ...
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Help with simplifying this integral
Is it possible to convert this expression:
$$\int u^2 \, f''(u) \, du$$
Into some integral of this form:
$$\int t^n \, f^{(n+1)}(t) \, dt$$
Using multiple integration techniques repeatedly like ...
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Context-Free Grammars and the Substitution of Equivalent Non-Terminal Symbols
Let’s say we have a Context-Free Grammar (CFG) $G=(N, \Sigma, P, S)$ where $L_G(X) = \{\, w \in \Sigma^* \mid X \Rightarrow^* w \,\}$ is the language of the non-terminal symbol $X \in N$.
I am ...
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Prove that: $\frac{a}{\sqrt{7a^2+36ab+bc+ca}}+\frac{b}{\sqrt{7b^2+36bc+ca+ab}}+\frac{c}{\sqrt{7c^2+36ca+ab+bc}}\le \frac{2}{\sqrt{13}}$
Let $a$, $b$ and $c$ be non-negative numbers such that $ab+ac+bc\neq0$. Prove that:
$$\frac{a}{\sqrt{7a^2+36ab+bc+ca}}+\frac{b}{\sqrt{7b^2+36bc+ca+ab}}+\frac{c}{\sqrt{7c^2+36ca+ab+bc}}\le \frac{2}{\...
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Reference request: Symbolic integration and Complex integration are consistent?
Background:
I'm studying symbolic integration and I've covered Liouville's theorem and its proof in class. I'm now trying to prove for example $\exp(\exp(x))$ is not elementarily integrable.
The ...
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How to solve generalization of inequality problem using substitution?
Let $a, b, c$ be nonnegative real numbers that sum to $1$. Prove that $$\sqrt{a+\frac14(b-c)^2}+\sqrt b + \sqrt c \le \sqrt3.$$
To solve this problem we use the substitution $2x = \sqrt b+\sqrt c$ ...
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Prove $2(ab+bc+ca)+\sqrt{a}+\sqrt{b}+\sqrt{c}\ge \sum \sqrt{a+4b+4c},\forall a,b,c,>0:abc=1$
I'm looking for some ideas to solve the following inequality.
Problem. Let $a,b,c$ be positve real numbers with $abc=1.$ Prove that$$2(ab+bc+ca)+\sqrt{a}+\sqrt{b}+\sqrt{c}\ge\sqrt{a+4b+4c}+\sqrt{b+...
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3 variables cyclic rearrangement inequality.
For positive $a, b, c$, prove that $$\sum_{\text{cyc}}\frac{a^2b(b-c)}{a+b} \geq 0.$$
The LHS being symmetric, we can assume $a \ge b \ge c$. I tried combining the denominators ; since $(a+b)(b+c)(c+...
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Prove $\sum\frac{1}{\sqrt{3a^2+4}} \ge 1, \forall a,b,c \ge 0:ab+bc+ca+abc=4$
I'm looking for some ideas to solve the following inequality.
Problem. Let $a,b,c\ge 0: ab+bc+ca+abc=4$ then prove$$\color{black}{\frac{1}{\sqrt{3a^2+4}}+\frac{1}{\sqrt{3b^2+4}}+\frac{1}{\sqrt{3c^2+...
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Prove $\frac14<\frac{a+b}{a+b+c}\times\frac{b+c}{a+b+c}\times\frac{c+a}{a+b+c}\le\frac8{27}$
The question:
If $a$, $b$ and $c$ are the sides of a triangle, prove that:
$\frac{1}{4}<\frac{a+b}{a+b+c} \times \frac{b+c}{a+b+c} \times \frac{c+a}{a+b+c} \leq \frac{8}{27}$
My approach:
I have ...
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Prove $\sqrt{(a+1)(b+1)}+\sqrt{(b+1)(c+1)}+\sqrt{(c+1)(a+1)} \leq 2\sqrt{ab+bc+ca+a+b+c+3}.$
Let $a,b,c>0$ and $abc=1$. Prove that: $$
\sqrt{(a+1)(b+1)}+\sqrt{(b+1)(c+1)}+\sqrt{(c+1)(a+1)} \leq
2\sqrt{ab+bc+ca+a+b+c+3} $$
What I have tried:
\begin{align}
& \text{Let } x=a+1,\;y=b+1,...
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How to properly use substitution in first-order logic? [duplicate]
Let's say I want to substitute $x$ for $y-1$ in statement $(\forall x)(x>0)$. So I should get $(\forall y)(y>1)$.
What is the logical basis for getting the result?
It seems that I need to prove ...
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Why can't I calculate the indefinite integral of $(5-x^2)^{-3/2}$ over the interval $[-1;2]$ using another substitution instead of $x=\sqrt5\sin(u)$?
I have been preparing for my university mathematical analysis course's exam and grinding away integrals amidst the other exercises on the syllabus. I have been stuck on one specific question regarding ...
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