Questions tagged [substitution]
Questions that involve a replacement of variable(s) in an expression or a formula.
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Trouble with such u-substitution that $du=e^{-2\pi i f(t)}f'(t)dt$ for $I=\int_0^1e^{-2\pi i f(t)}dt$ and integration by parts
Update: After depressingly many hours I realized that the task of bounding the integral is quite simple and in all likelihood, there is no such $u$-substitution conundrum as I have discussed here (you ...
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3
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How to prove $\sqrt{(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)}\ge 3+\frac{1}{3}\sum_{cyc}\left(\frac{b-c}{b+c}\right)^2.$
If $a,b,c>0$ prove that $$\sqrt{(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)}\ge 3+\frac{1}{3}\sum_{cyc}\left(\frac{b-c}{b+c}\right)^2.$$
It is stronger than the well-known result $$(a+b+...
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Question on substitution
The Wikipedia article on substitution states:
In first-order logic, a substitution is a total mapping $σ: V → T$ from variables to terms; many, but not all authors additionally require $σ(x) = x$ for ...
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Trigonometric substitution in $\int \sqrt{\frac{x+7}{x}}\ dx$
For a linear term in integral like
$$\int\sqrt{\frac{x+7}{x}} \ dx$$
can we substitute $x = 7\tan^2(\theta)$ to solve this integral?
3
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6
answers
130
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Trouble understanding this hard integral
here is the integral:
$$\int \frac{x^3}{(x^2+1)^2}dx$$
now I was trying $U = x^2+1 $ then $du = 2xdx$:
$$\int \frac{x^3}{u^2} \frac{du}{2x}$$
$$ \frac{1}{2}\int \frac{x^2}{u^2} du$$
How can I continue ...
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Cauchy Principal value and Substitution
In general we cannot do substituion when we are dealing with Cauchy principal value integrals. Take for instance the function $\varphi(x)= x^3$ for $x\le0$ and $\varphi(x)= x^2$ for $x\ge0$ and ...
1
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2
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Integrating $\int\frac{\cos^2x}{\left(\sin^2x+4\cos^2x\right)^2} \, dx $ [closed]
I'm looking at this integral as part of a problem set for a class. So far, we've been given two hints: we should use $x=\arctan(t)$ and we should try dividing the numerator and denominator by $\cos(x)$...
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How to prove $ a+b+c+\sqrt{bc}+\sqrt{ca}+\sqrt{ab}\ge 6$?
Question. Prove $$ a+b+c+\sqrt{bc}+\sqrt{ca}+\sqrt{ab}\ge 6,$$ when $a,b,c\ge 0: ab+bc+ca+abc=4.$
My idea:
I've tried to use AM-GM as
$$\bullet \sum \sqrt{ab}\ge 2\sum \frac{ab}{a+b}=2(ab+bc+ca)\sum \...
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Showing $\sum_{i=1}^n\tan\alpha_i\geq (n-1)\cdot \sum_{i=1}^n\cot\alpha_i$, for real $\alpha_i\in(0,\pi/2)$ with $\sum_{i=1}^n\cos^2\alpha_i=1$
Real numbers $\alpha_1,\ldots,\alpha_n \in \left(0,\ \frac{\pi} 2\right)$ satisfy the condition $\sum_{i=1}^n\cos^2\alpha_i=1$.
Prove that $$\sum_{i=1}^n\tan\alpha_i\geq (n-1)\cdot \sum_{i=1}^n\cot\...
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$\int\frac{e^x}{\sqrt{1-e^{2x}}}\,dx$ with $e^x=\cos(u)$
Consider the integral
$$
I=\int\frac{e^x}{\sqrt{1-e^{2x}}}\,dx
$$
When using the substitution $e^x=\sin(u)$, I get $I=\arcsin(e^x)+C$. However, when using $e^x=\cos(u)$, I get $du/dx=-e^x/\sin(u)$ and ...
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Inequality $\frac{a}{\sqrt{b}} + \frac{b}{\sqrt{a}} \ge \sqrt[4]{8}$ on the unit circle
While playing with a few two variable inequality and AM-GM inequality, I have ran into the following puzzle:
Question: Show that if $a, b \in (0,1)$ and $a^2+b^2 = 1$, then: $\dfrac{a}{\sqrt{b}} + \...
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Why the following transformation true?
$\int \sec x^2\tan x\mathrm{d}x$=$\int u'u\mathrm{d}u$
The eqution comes from a answer to MIT18.01 problem sets. I know that 'u' is the previous 'tanx', but I don't understand why 'dx' turns into 'du'...
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Why are the following two integrals correct? [closed]
$\int \frac{1}{(1-x)^2}\mathrm{d}x$=$\frac{1}{1-x}$
$\int \frac{1}{(1-x)^2}\mathrm{d}x$=$\frac{x}{1-x}$
This question comes from MIT18.01 problem sets. I think that a constant is missing from the ...
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Roots of $ 16 x^5 - 20 x^3 + 5x + 1 = 0 $
The following is from Edexcel further mathematics Core Pure Book 2 A Level Mixed Exercise 1 Question 9 part b:
9 a Use De Moivre's Theorem to show that
$$ \cos 5\theta \equiv 16 \cos^5 \theta - 20\...
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Given $x>0,y>0$, AND $\frac{3}{2 x^2+3 xy}+\frac{5}{3 xy+4 y^2} =2$, So max{xy}?
Given $x>0;y>0$; if $\frac{3}{2 x^2+3 xy}+\frac{5}{3 xy+4 y^2} =2$, Find max{xy}?
Here is my try:
Solution 1:
$xy=t$,
$\frac{3}{2 x^2+3t}+\frac{5}{3t+4y^2} =2$
$y = \frac{t}{x}$
$\frac{3}{2 x^...
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Prove $\sqrt{\frac{a+4bc}{4a+bc}}+\sqrt{\frac{b+4ca}{4b+ac}}+\sqrt{\frac{c+4ab}{4c+ab}}\ge 3,$ when $a+b+c=3.$
Problem. If $a,b,c\ge 0: ab+bc+ca>0$ and $a+b+c=3,$ prove that$$\sqrt{\frac{a+4bc}{4a+bc}}+\sqrt{\frac{b+4ca}{4b+ac}}+\sqrt{\frac{c+4ab}{4c+ab}}\ge 3.$$
It was here.
Equality holds at $a=b=c=1$ ...
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Need help changing the bounds for an integral involving the Jacobian
Use the transformation $u = x + 2y$, $v = y-x$ to evaluate $\displaystyle \int_{0}^{\frac{2}{3}} \int_{y}^{2-2y}\left(x+2y\right)e^{y-x} \, dx \, dy$.
I started with calculating the jacobian:
$J(u,v) =...
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Integration: How to generally pick "$u$" in substitution and some specific examples I had to do [duplicate]
I have been doing integration by substitution. And I wonder whether there is a general method or approach, as to what I can pick as $u$. I have had these specific tasks, were the teacher gave a hint ...
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Limit substitution domain restriction
I apologize in advance for the imprecise question but I don't really know what I am looking for here.
In doing classwork I have come across various limits where the obvious substitution would be $t=x^...
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Stuck at understanding how to properly use the change of variables theorem when localizing the smooth bump function to a given open ball
This is question is spin-off of my prior question: Trouble with change of variables when constructing a smooth bump function localized on a given open ball, but this one is more suited to ...
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Trouble with change of variables when constructing a smooth bump function localized on a given open ball
I seem to have lost the ability to do u-substitutions as the following question will show: I am trying to construct a smooth compactly supported function such that it is equal to one on a given open ...
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$ \int {(1-x^2)e^{-\frac{x^2}{2}} dx} $
$ \int {(1-x^2)e^{-\frac{x^2}{2}} dx} $
My try,
substituing, $e^{-\frac{x^2}{2}}=y$ but this doesnot work.
But, came to know it has very good answer: $xe^{-\frac{x^2}{2}}+c$.
So, if I would ...
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How was this substituted?
A while ago, I saw a post for a solution of the time of the brachistochrone. Sadly, I do not get how the integral shown was substituted. Can anybody help?
The original integral was
$$T=\sqrt{\frac{1}{...
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Solving $\int \frac{3}{2} x e^{-6 x} d x$ using $u$-substitution.
$$\int \frac{3}{2} x e^{-6 x} d x$$
Hi, I was solving an equation and I came to a part requiring that I solve $\int{\frac{3}{2}xe^{-6x}dx}$.
I attempted to solve this using $u$-sub but got the wrong ...
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Functional Equation Solution?
I've been trying to solve this:
$f: \Bbb{R\setminus \{1\} \to \Bbb{R}}, f(2-x) + x f(\frac{1}{1-x}) = 3-x $
Seems sensible to simplify slightly by replacing $x$ with $(2-x)$ to give:
$f(x) + (2-x)f(\...
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Where did I go wrong in solving this integral of Bessel functions using the Feynman technique?
While solving a static problem regarding an isotropic cylinder under its own weight, I ran into the following integral:
$$ I_0 = \int_0^a r^2 J_0(\mu_ir) J_1(\mu_jr)dr \tag 1$$
where $r$ represents ...
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Trigonometric Substitution Definition
Trigonometric substitution is defined as the method of replacing variables of integration with trigonometric functions. What I don’t understand is how is it that in the hypotenuse of a right triangle, ...
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Why can't I solve $\int \sin^3(x)dx$ directly using u-substitution without trig identities?
I saw that the standard solution to the problem
$\int \sin^3(x)dx$
involves first using trig identities and then u-substitution. I am not understanding why it is wrong to use directly u-substitution, ...
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Hey can someone help understanding this topic! U sub versus no U sub!
I want to understand how these two integrals are equal
say we integrate $$\int(x-10)dx$$
if we do it normally we will get $$\frac12x^2-10x$$
however we can also u-sub to save time $$\int u du$$ where ...
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$\Gamma (x) \cos(ax)$ identity
I am asked to show that $$\Gamma (x) \cos(ax) = b^x \int_{0}^{\infty} \mathrm{d} t \enspace t^{x-1} e^{-bt \cos(a)} \cos(bt \sin(a)).$$
A change of variables $t \to \frac{t}{b}$ shows that $b$ is ...
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How to calculate the first order correction to the asymptotic solution to a second order differential equation
Consider the following differential equation:
$$
\frac{d^{2}y}{dx^{2}}=\frac{1}{2}\begin{cases}
1-e^{-\frac{y}{\epsilon}},\space\space\space x<0\\
e^{-\frac{1-y}{\epsilon}}-1,\space\space\space x&...
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Solving $T(n) = T(n/2) + T(n/3) + n$
Show that the solution to the recurrence relation
$$T(n) = n \;\;\;\;\text{ for }\; n=1,2$$
$$T(n) = T(n/2) + T(n/3) + n \;\;\;\;\text{ for }\; n > 2$$
is $O(n)$ using substitution.
$$T(n) \leq c\...
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Solving a polynomial equation using substitution
The equation is:
$x^4 + 4x^2 + 16 = 0$
I tried solving it by substitution and then using the quadratic formula:
$x^2 = a$
$a^2 + 4a + 16 = 0$
using the quadratic formula I got $a = -2 \pm 2i\sqrt{3}$,
...
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$u$-substitute in integration
So the $u$-substitute theorem states that $$\int f(g(x))g’(x)~dx = \int f(u)~du$$
But when we use the $u$-substitution, we say $u=g(x)$ right?
So my question is why when we use this we can change $u$ ...
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Verification of solution by $u$-substitution of $\int \frac{\ln x}{1+x^2}dx$
I was trying to solve the integral $\int \frac{\ln x}{1+x^2}dx$.
using the sub $u=\frac{1}{x}$ which means $x= \frac{1}{u}$ and $dx=\frac{-du}{u^2}$, so the integral becomes
$$\int \frac{\ln\frac{1}{u}...
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Completing the square under a square root
Trying to work through a problem that requires us to complete the square, then use a sinh substitution...but I need to start by remembering back to math from many years ago.
$$\int\sqrt{2x^2+3x+4} \ ...
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3
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Why does sometimes the value of definite integral becomes $0$ for a non zero graph of a function
For some arbitrary constant $a$, the question is: $$\int_{0}^{ \pi} \frac{x}{a^2- \cos^2(x)} dx$$
I was able to bring it down to:
$$I=\frac{\pi}{2}\int_{0}^{ \pi} \frac{1}{a^2- \cos^2(x)} dx$$
I tried ...
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Properties of substitution system
I'm interested in automatic sequence or substitution systems.
I focused on the simplest form with a binary alphabet without constants.
A well-known example is Thue-Morse sequence
with axiom ...
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Regarding improper integral and normal distribution
I am trying to find the expectation of the normal distribution with parameters $\mu$ and $\sigma^2$. My methodology is to show that $E(X-\mu)$=0.
So I begin calculating the integral $\frac{1}{\sqrt{2\...
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using induction to prove $2^n \geq n + 5$
When using induction to prove $P(n) : 2^n \geq n + 5$ for $n \geq 3$ there's one part of the induction step I'm really struggling with:
Assuming that $P(k) : 2^k \geq k + 5$ , we want to prove $P(k + ...
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3
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Integrate $\frac{x}{\sqrt{1-x^2}}$
I want to integrate
\begin{equation*}
\int_{a}^b\frac{x}{\sqrt{1-x^2}}dx
\end{equation*}
by using the substition $\varphi(t)=\sqrt{1-x^2}$. For sake of simplicity we assume $0< a< b< 1$. If ...
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Alternate method to solve this integral [closed]
$$
\int \! \sqrt{\frac{x}{x^3 - a^3}} \, dx
$$
I solved this integral by substitution method by taking $t= x^{3/2}$. Is there any other method to solve this problem?
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If $a^2+b^2+c^2+abc=4$, Find minimum $P=\sqrt{\frac{2a+bc}{3}}+\sqrt{\frac{2b+ca}{3}}+\sqrt{\frac{2c+ab}{3}}-\frac{3(a+b+c+abc)}{2}$
Let $a,b,c\ge 0: a^2+b^2+c^2+abc=4$. Find minimum $$P=\sqrt{\frac{2a+bc}{3}}+\sqrt{\frac{2b+ca}{3}}+\sqrt{\frac{2c+ab}{3}}-\frac{3(a+b+c+abc)}{2}$$
When $a=b=c=1,$ we get that $P\ge -3$ So we need to ...
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How to prove $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{16(ab+bc+ca)^2}\ge \frac{1}{a+b+c}.\left(\dfrac{1}{\sqrt{abc}}-3\right)$
If $a,b,c$ are postive real numbers. Prove that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{16(ab+bc+ca)^2}\ge \frac{1}{a+b+c}.\left(\dfrac{1}{\sqrt{abc}}-3\right).$$
My trying based on a guess ...
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2
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Prove $\sum_{cyc}xy\left(2\sqrt{z^2+1}-z\right)\ge\sum_{cyc}\left(\sqrt{z^2+1}-z\right)$ when $xy+yz+zx=1$
Given $x,y,z$ be non negative real numbers satisfying $xy+yz+zx=1.$ Prove that
$$\sum_{cyc}xy\left(2\sqrt{z^2+1}-z\right)\ge\sum_{cyc}\left(\sqrt{z^2+1}-z\right). $$
My thoughts is proving$$xy\left(2\...
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votes
3
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Regular Season Problem 11 from 2023 MIT Integration Bee
$$
\int \left(\sqrt{2\log x}+ \frac{1}{\sqrt{2\log x}} \right) dx
$$
I am stuck on this problem from This years integration bee. I have tried substitution but it is not giving the correct answer which ...
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vote
0
answers
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Rescaled solution of the PDE Vlasov equation are again solutions
In order to use the advantages of rescaling, I want to show that a rescaled solution of the Vlasov equation is again a solution. The setting is a gravitational Vlasov-Poisson system, which is a system ...
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Natural Log Functions Integration [closed]
I've found that $\ln(|\tan(x)|)$ is the solution.
I need help evaluating this problem.
Attempts:
$du = (-\csc(x)\cot(x)-\csc^2(x))dx$
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Evaluating an integral with a $\sqrt{x^2+1}$ in it ; How to solve it?
Problem:
Perform the following integration:
$$ \int \dfrac{\sqrt{x^2+1}}{x} \,\, dx $$
Answer:
Let $I$ be the integral we are trying to evaluate.
\begin{align*}
I &= \int \sqrt{ 1 + \dfrac{1}{x^2} ...
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0
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Substitution in this laplace integral, how the orignal complex variable becomes a real variable
I'm learning cagniard de-hoop method(de Hoop, A. T., A modification of Cagniard’s method for solving seismic pulse problems, Appl. Sci. Res. B 8, 349-356 (1960). ZBL0100.44208.), but I don't quit ...
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