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Questions tagged [substitution]

Questions that involve a replacement of variable(s) in an expression or a formula.

0
votes
2answers
33 views

Differential equation substitution f(x/t)

Hello I have the differential equation $$x'=\frac{8t+10x}{17t+x}$$ Brought it in a eqation where I can substitute $$ u=\frac{x}{t}$$ and after some transformations I got the equation $$ \frac{17+u}{8-...
1
vote
6answers
95 views

real solution of equation $(x^2+6x+7)^2+6(x^2+6x+7)+7=x$ is

Number of real solution of equation $(x^2+6x+7)^2+6(x^2+6x+7)+7=x$ is Plan Put $x^2+6x+7=f(x)$. Then i have $f(f(x))=x$ For $f(x)=x$ $x^2+5x+7=0$ no real value of $x$ For $f(x)=-x$ $x^2+8x+7=0$...
0
votes
1answer
40 views

How to use half angle trigonometric substitution on x/cos(x)

I have this integral that I want to solve for homework: $\int \frac{x}{\cos\left(x\right)}\mathrm{d}x$. After some research, I found out that I can use a half-angle substitution to solve similar ...
0
votes
2answers
17 views

$U = \frac{h\omega}{e^{hw/KT}-1}\approx KT - \frac{h\omega}{2}+…O(\frac{h\omega}{KT})$

$$U = \frac{h\omega}{e^{hw/KT}-1}\approx KT - \frac{h\omega}{2}+....O(\frac{h\omega}{KT})$$ If have to prove this for $KT\gg h\omega$ I dont understand what the O in the equation means. can someone ...
1
vote
2answers
45 views

The integral of $5/\left(x^2+2\right)$

I have to calculate a integral for following equation: $\frac{5}{x^2+2}$. On the integral calculator they show that it must be solved by substitution and the substitution must be $u=\frac{x}{\sqrt{2}}$...
1
vote
0answers
65 views

Change of variables, why does integral disappear?

In a PDF about Gaussian-Stochastic models for spectral diffusion, there is a simplification made by change of variable between equations 7.21 and 7.22 that I just can't seem to make. Writing it here, ...
35
votes
6answers
4k views

Why is this integration method not valid?

Let $$I=\int \frac{\sin x}{\cos x + \sin x}\ dx \tag{1}$$ Now let $$u=\frac{\pi}{2} - x \tag{2}$$ so $$I=\int \frac{\sin (\frac{\pi}{2} - u)}{\cos (\frac{\pi}{2} - u)+\sin (\frac{\pi}{2} - u)}\ du \...
0
votes
0answers
58 views

Prove $\sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,a+ b\,)}}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,c+ a\,)}}$ with $a,\,b,\,c> 0$

Let $a,\,b,\,c$ be positive numbers. Prove that $$\sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,a+ b\,)}}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,c+ a\,)}}$$ I tried Holder and $\lceil$ https://...
-2
votes
3answers
86 views

Problem with integrating $\int_0^{\pi/2}\frac{\cos^6x}{\cos^6x+\sin^6x}dx$ [duplicate]

Someone told me there is an equation $$\int_0^{\pi/2}f(sinx)dx=\int_0^{\pi/2}f(cosx)dx$$ With this equation, it's easy to get the answer$\frac{\pi}{4}$. What I want to know is why we have this ...
0
votes
0answers
63 views

Prove that $\sum\limits_{cyc}\,\frac{a^{\,2}}{bc+ a}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{2\,bc+ 2}}$ [on hold]

Let $a,\,b,\,c$ be positive numbers. Prove that $$\sum\limits_{cyc}\,\frac{a^{\,2}}{bc+ a}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{2\,bc+ 2}}$$ I tried Holder Inequality (it's only the hint to get you ...
3
votes
2answers
46 views

Why can we let $x = 2\cos t\ $ in the solution for the following system of equations

Solve in real number the system of equations $\begin{cases}x^2 = y+2 \\ y^2 = z+2 \\ z^2 = x+2 \end{cases}$ The solution given to me says the following: If we eliminate $y$ and $z$, we obtain a ...
10
votes
3answers
437 views

Evaluating $\int_0^1 \frac{3x}{\sqrt{4-3x}} dx$

So this is the integral I must evaluate: $$\int_0^1 \frac{3x}{\sqrt{4-3x}} dx$$ I have this already evaluated but I don't understand one of the steps in its transformation. I understand how ...
0
votes
3answers
92 views

How to find $f(X)$ such that $\sum\limits_{cyc}a^2-f(X)[abc-(1-a)(1-b)(1-c)]\geqq\frac{3}{4}X^2$ for $abc=(X- a)(X- b)(X- c),0\leqq a,\,b,\,c\leqq X$?

We have $\sum\limits_{cyc}\,a^{\,2}\geqq \frac{3}{4}\,X^{\,2}\tag{HaiDangel29}$ with $abc= (\,X- a\,)(\,X- b\,)(\,X- c\,),\,0\leqq a,\,b,\,c\leqq X$. Here is a hint to get you started from above. For $...
2
votes
2answers
82 views

Find maximize of the function $\frac{a}{1+a^2}+\frac{b}{1+b^2}-\frac{1}{c^2+1}$

Let $a,b\in R^+$ such that $ab+bc+ca=1$. Find the maximize of $$P=\frac{a}{1+a^2}+\frac{b}{1+b^2}-\frac{1}{c^2+1}$$ By Wolframalpha i can see that if $a=b=2-\sqrt 3;c=\sqrt 3$ we will have $P=\dfrac ...
-1
votes
2answers
24 views

Using substitution for definite integral [closed]

I chose E for this question, but the answer is C. Why does the interval changes after the substitution? I first calculate $du = 2x \space dx$ and then substitute $u$ and $du$ into the expression and ...
0
votes
1answer
57 views

Prove $k=0$ is the only non-negative $k$ such that $\sum\limits_{cyc}\,a^{\,3}- \sum\limits_{cyc}\,a^{\,2}b\geqq k(\,a- b\,)(\,a- c\,)(\,b+ c\,)$

Prove with $a+ b,\,b+ c,\,c+ a\geqq 0$ $$\begin{equation}\begin{split} k= constant= 0 \end{split}\end{equation}$$ is the only non-negative $k$ such that $$\begin{equation}\begin{split} \sum\limits_{...
0
votes
1answer
83 views

Prove $\frac{1}{a+ 2\,b}+ \frac{1}{b+ 2\,c}+ \frac{1}{c+ 2\,a}\leqq 1$ with $8\,abc\geqq a+ b+ c+ 5$ and $a,\,b,\,c> 0$

Prove $$\frac{1}{a+ 2\,b}+ \frac{1}{b+ 2\,c}+ \frac{1}{c+ 2\,a}\leqq 1$$ with $8\,abc\geqq a+ b+ c+ 5$ and $a,\,b,\,c> 0$ $$constant= 8$$ is the best $constant$, which was found by me (using ...
0
votes
2answers
27 views

Maclaurin series degree 6 involving substitution

So I am aware that you can perform operations on taylor series, such as integration, differentiation, etc. However, I am not sure of when exactly one is allowed to substitute values into another ...
0
votes
1answer
95 views

Prove $\prod\,\left ( a+ \frac{1}{a} \right )- \frac{4}{3}\sum\,\frac{b+ c}{a}\geqq 0 $

Prove $$\begin{equation}\begin{split} \prod\,\left ( a+ \frac{1}{a} \right )- \frac{4}{3}\sum\,\frac{b+ c}{a}\geqq 0 \end{split}\end{equation}$$ with $a,\,b,\,c> 0$. $$\begin{equation}\begin{...
0
votes
1answer
17 views

Question about using substitution + Jacobian for integration in a concrete example

I am new to calculus and was hoping for some feedback re the following question and my proposed answer. Many thanks in advance. Parallelogram $D$ in the first quadrant has corners at $(0,0)$, $(2,2)...
2
votes
1answer
89 views

show this $\sum_{cyc}\frac{x}{x^2-x+1}\le\frac{8}{3}$ [duplicate]

let $x,y,z,w\in R$,and such $x+y+z+w=2$.show that $$\sum_{cyc}\dfrac{x}{x^2-x+1}\le\dfrac{8}{3}$$ I have only solve when $x,y,z,w>0$, because $$\dfrac{x}{x^2-x+1}\le\dfrac{4}{3}x$$ so $$\sum_{...
2
votes
1answer
65 views

Definite integral containing 2 trig functions and a square root function

$$\int_{-\pi/4}^{\pi/4}\bigl(\cos x+ \sqrt {1+x^2}\sin^3x \cos^3x \bigr)\, dx $$ This question is from a math GRE practice test I've tried to solve this integral for 2 days... starting to think it ...
0
votes
1answer
217 views

Inequality for $a,b,c>0$ $\sum_{cyc}\sqrt{\frac{a^3}{14a^2+4b^2}}\leq \sum_{cyc}\sqrt{\frac{a+b}{36}}$

A friend gives me the following result : Let $a,b,c>0$ then we have : $$\sqrt{\frac{a^3}{14a^2+4b^2}}+\sqrt{\frac{b^3}{14b^2+4c^2}}+\sqrt{\frac{c^3}{14c^2+4a^2}}\leq \sqrt{\frac{a+b}{36}}+\...
1
vote
2answers
91 views

Find maximum of function $A=\sum _{cyc}\frac{1}{a^2+2}$

Let $a,b,c\in R^+$ such that $ab+bc+ca=1$. Find the maximum value of $$A=\frac{1}{a^2+2}+\frac{1}{b^2+2}+\frac{1}{c^2+2}$$ I will prove $A\le \dfrac{9}{7}$ and the equality occurs when $a=b=c=\dfrac{...
2
votes
1answer
74 views

A hard inequality $(a^2-ab+b^2 )(b^2-bc+c^2 )(c^2-ca+a^2 ) + 11abc \leq 12$

Given $$c=\min⁡(a,b,c)~, \quad a+b+c=3 \\ P=(a^2-ab+b^2 )(b^2-bc+c^2 )(c^2-ca+a^2 )~,$$ I have to prove that $$P+11abc \le 12~.$$ I started with $$b^2-bc+c^2 \le b^2 \quad \text{and} \quad c^2-ca+...
4
votes
1answer
97 views

Proving $ \sum_{cyc}^{} \frac {a(a^3+b^3)}{a^2+ab+b^2} \ge \frac{2}{3} (a^2+b^2+c^2)$ for $a, b, c > 0$

For $a,b,c>0$, I have to prove that $$ \sum_{cyc}^{} \frac {a(a^3+b^3)}{a^2+ab+b^2} \ge \frac{2}{3} (a^2+b^2+c^2).$$ We have: $$\begin{align} \sum_{cyc}^{} \frac {a(a^3+b^3)}{a^2+ab+b^2} &= \...
4
votes
4answers
127 views

How to solve $\int^1_{-1} \frac{\sin(x)}{1+x^2}dx$?

I have to solve $$\int^1_{-1} \frac{\sin(x)}{1+x^2}\,dx$$ I am a Calculus 1 student, and I am having difficulty because I can't think of anything that I could make into a substitute which would ...
-1
votes
1answer
40 views

why is the following U-Substitution wrong?

It is known that $$ \int \frac{1}{\sqrt{1-x^2}} dx = arcsin(x)+c$$ this can be done utilizing u-substitution $ x = sin(u) $ However, i can let $ u = 1-x^2 $ $dx = -2u \, du $ which gives ...
0
votes
2answers
93 views

Prove/disprove $\sum_{cyc}a\sqrt{\frac{(ca + 1)(ab + 1)}{bc + 1}} \ge 2$ where $a$, $b$, $c > 0$ and $a^2 + b^2 + c^2 = 1$

$a$, $b$ and $c$ are positives such that $a^2 + b^2 + c^2 = 1$. Prove/disprove that $$a\sqrt{\frac{(ca + 1)(ab + 1)}{bc + 1}} + b\sqrt{\frac{(ab + 1)(bc + 1)}{ca + 1}} + c\sqrt{\frac{(bc + 1)(ca + 1)}{...
3
votes
3answers
119 views

How do i integrate $\int \frac{dx}{(2x+3)\sqrt{(x^2+3x+2})}$?

Integrate $\int \frac{dx}{(2x+3)\sqrt{(x^2+3x+2})}$ I put $x^2+3x+2=t,$ and notice that $2x+3 dx=dt$, but the $dx$ is above! Please help me!
1
vote
3answers
92 views

Prove that $\sum_{cyc}\frac{a}{b^2}\ge 3\sum_{cyc}\frac{1}{a^2}$ for $a,b,c>0$ such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$

$a$, $b$ and $c$ are three positives such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = 1$. Prove that $$\dfrac{a}{b^2} + \frac{b}{c^2} + \frac{c}{a^2} \ge 3 \cdot \left(\frac{1}{a^2} + \frac{1}{...
1
vote
2answers
92 views

Value of $\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}}+\frac{1}{\sqrt{1+c^2}}+\frac{1}{\sqrt{1+d^2}}$

If $$\frac{a+b+c+d}{\sqrt{(1+a^2)(1+b^2)(1+c^2)(1+d^2)}}= \frac{3\sqrt{3}}{4}$$ for $a,b,c,d>0$ Then Value of $\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}}+\frac{1}{\sqrt{1+c^2}}+\frac{1}{...
0
votes
1answer
20 views

How can I determine the form of a geometric series in a recurrence relation by using iterative substitution?

This question is based more in computer science however I think a mathematics approach is probably better. When solving a recurrence relation using iterative substitution you generally need to find ...
7
votes
1answer
122 views

Antiderivative of $(1+x^3)^{-1/3}$.

I tried multiple substitutions while trying to solve for: $$ \int\frac{{\rm d}x}{(1+x^3)^{\frac{1}{3}}} $$ According to some online sources it is non-elementary, although the question was taken from ...
0
votes
1answer
34 views

Solving Higher Order Differential Equations

I have to solve the following differential equation using the substitution $$u = y'$$ $$yy''+(y')^2 + 1 = 0 $$ But how do I integrate after simplifying the substitution if I cannot separate the ...
1
vote
1answer
75 views

Prove $F(\,k\,)=(\,16\,X^{\,}- 24\,X+ 18\,)k^{\,2}- 11\,X+ 1\geqq 0$

Given that $1\leqq X\leqq k,$ prove that $$F(\,k\,)=(\,16\,X^{\,}- 24\,X+ 18\,)k^{\,2}- 11\,X+ 1\geqq 0 \tag{29}$$ Origin For $a,\,b,\,c\geqq 0$ and $a+ b+ c= 3$, prove that $(2+a^2)(2+b^2)(2+c^2)+...
0
votes
0answers
34 views

Most General unifier in logic

i have a question about most general unifier in logic. i'll begin by saying that in the class we were only given a summary in a few words, without any example, and they just moved on to the next topic ...
1
vote
2answers
51 views

trouble in solving $\int\frac{1}{t(t+2)} dt$ by using a specific variable substitution

Before anything else, I would like to apologize for my english as it is not my mother tongue and I may use it unperfectly. I would like to solve $\int_a^b\frac{1}{t(t+2)}dt$. Of course, it is doable ...
0
votes
1answer
25 views

How does doing a U- Substitution come out with this outcome?

I saw this on a website. I am working on Integrals and the U-Substitution didn't make sense. The original problem was $\int$x/x+1$dx$ The U-Sub did u = x+1 , du = dx then it gave me a $\int$u-1/u ...
0
votes
0answers
34 views

Why can you use substitution to find a polynomial with roots g(a), g(b)

In a maths textbook is this statement which pertains to roots of polynomials: If an equation in x has a root x=p, and if we make a substitution u = f(x), then the resulting equation in u has a root ...
1
vote
1answer
75 views

Finding a better bound in an inequality [closed]

Consider points $(x,y)$ on the curve $\sqrt{x^2-3x}+\sqrt{y^2-3y}=1$. Prove that for all such pairs: $$x^2+y^2\lt2(x+y)+8.$$ NOTE.- This problem was proposed by two mathematicians, from Romania and ...
2
votes
3answers
67 views

What Are The Steps to Evaluate the Integral $\int_{0}^{5}s\sqrt{25-s^2}$ With u-substitution?

Summary Forgive me. I'm brand new to integrals and I feel like we went from 0 to 100 really fast in class. I'm sure I'm missing a step somewhere or have a calculation that's off. Problem Use the ...
1
vote
1answer
24 views

greatest value of function depends on parameter $k$

Find the greatest value of the function $f(x)=x^4-6kx^2+k^2$on the interval $[-2,1]$ depending on the parameter $k$ My Try: $$f(x)=x^4-6kx^2+9k^2-8k^2$$ $$f(x)=(x^2-k)^2-8k^2$$ from $-2 \...
0
votes
2answers
93 views

$\frac{\left(10^4\right)}{x^2}=\frac{\left(x^{\left(8-2\log x\right)}\right)}{10^4}$ Solve for x.

$\frac{\left(10^4\right)}{x^2}=\frac{\left(x^{\left(8-2\log x\right)}\right)}{10^4}$ Solve for x. $\frac{\left(10^4\right)}{x^2}=\frac{\left(x^{\left(8-2\log x\right)}\right)}{10^4}\Rightarrow 10^8=\...
1
vote
3answers
59 views

$\int \frac{1}{{1-2x-x^2}} \, \mathrm{d}x $ substitution

I have this integral. $$\int \frac{1}{{1-2x-x^2}} \, \mathrm{d}x $$ But I am unable to do it right and I just don't know where is the problem in my steps. My steps: Complete the square $$\int \...
0
votes
2answers
37 views

Substitution in easy definite integral

Let me ask you the following very easy question, but I have some problems. I want to show that $$\int \limits_{0}^{2\pi}\dfrac{d\theta}{a+\cos \theta}=2\int \limits_{0}^{\pi}\dfrac{d\theta}{a+\cos \...
2
votes
2answers
74 views

Let $P (x )=x^4+ax^3+bx+c=0$ and have real coefficient and have all real roots . Prove that $ab \leq 0$

Let $P (x )=x^4+ax^3+bx+c=0$ and have real coefficient and have all real roots . Prove that $ab \leq 0$ First Let the roots of this polynomial (call it P(x)) be $q,r,s,t$ By Vieta's, $a=-(q+r+s+t)$ ...
0
votes
0answers
161 views

Substitution and Transitive property in Logical Equivalence

I'm stuck in a discussion with a friend on some weekly tasks given to us and we're coming up with different opinions on this question that I'm hoping someone experienced might be able to expand on for ...
5
votes
2answers
70 views

Inequality related with $abcd=(1-a)(1-b)(1-c)(1-d)$ [duplicate]

Given that $0<a,b,c,d<1$ satisfying $abcd=(1-a)(1-b)(1-c)(1-d)$. Prove that $$(a+b+c+d)-(a+c)(b+d)\geq 1.$$ First, I have already done a quite similar exercise as below: "Given that $a^2+b^2+c^...
1
vote
1answer
51 views

Exercise XIX number 15 - Calculus Made Easy

$$ \text{Use substitution}\quad\frac{1}{x}=\frac{b}{a}cosh\;u\quad\text{to show that}\quad\\ \int\;\frac{dx}{x\sqrt{a^2-b^2x^2}}=\frac{1}{a}\ln\frac{a-\sqrt{a^2-b^2x^2}}{x}\;+\;C.\\ \text{My way:}\\ \...