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Questions tagged [substitution]

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

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2answers
18 views

Are these legitimate rules/formula for integration without using the substitution method?

I'm talking about: $\int(ax+b)^ndx=\frac{(ax+b)^{n+1}}{(n+1)(a)}$ $\int\frac{1}{ax+b}dx=\frac{1}{a}ln(ax+b)$ $\int e^{ax+b}dx=\frac{e^{ax+b}}{a}$ $\int a^{ax+b}dx=\frac{a^{ax+b}}{(ln|a|)(a)}$ Of ...
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0answers
11 views

Gaussian integral and polar change of variable

I would like to compute the Gaussian integral $\int_{-\infty}^{+\infty}e^{-x^2/2} dx$ using a polar substitution. I know the usual proof using Fubini's theorem and the polar substitution of variables....
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1answer
17 views

How to transform this Diffusion equation ?

I have a diffusion equation: $$ \nabla^2C = \frac{\partial C}{\partial t}$$. Now I would like to transform this equation into a co-ordiante frame which movies with the unperturbed palaner ...
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0answers
21 views

Integration of $g(x) = \begin{cases} x^{1/c-1}e^{-x^{-c}} &x>0\ \ \\ 0 &\text{else} \end{cases} \text{and} \ c\neq 0$

I need help to integrate this function: $$\begin{align} &g(x) = \begin{cases} x^{1/c-1}e^{-x^{-c}} \ &x>0\ \ \\ 0 &\text{else} \end{cases}\\[5pt] &\text{and} \ c\neq 0 \end{align}...
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1answer
17 views

Solve for x negative exponential

I have the following equation: $y = 14857x^{-1.092}$ I know $y = 43$, how do I rewrite this to solve for $x$. e.g. $43 = 14857x^{-1.092}$
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4answers
56 views

Methods to solve $\int_{0}^{\infty} x^{n}\cos(x)\:dx$

I've been playing around with the following integral and was wondering if it can be generalised to any Real $n$. Does anyone know of any methods to approach this one? $$ I = \int_{0}^{\infty} x^n \...
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2answers
57 views

Solving $\frac{dy}{d\theta} + y \cos \theta = \frac{1}{2} \sin 2\theta$

Solve $$\frac{dy}{d\theta} + y \cos \theta = \frac{1}{2} \sin 2\theta$$ with $y(\pi/2) = 4$. I couldn't think of any useful substitution, so I tried simplifying this DE, but didn't get anywhere. ...
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2answers
33 views

How can a doubly improper integral become a singly improper integral after substitution?

How can a doubly improper integral become a singly improper integral after substitution? If we take $x=\sin^2{t}$ then: $$\int_0^1\frac{1}{x\sqrt{1-x}}dx=2\int_0^{\pi/2}\csc{t}dt$$ On the lefthand ...
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4answers
35 views

solving an inseparable differential equation involving exponentials

Solve $2xe^y$ + $e^x$ + ($x^2$ + 1)$e^y$$\frac{dy}{dx}$ = 0 with $y$ = 0 when $x$ = 0. So this is clearly an inseparable differential equation so I thought the standard way to approach this was with ...
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3answers
45 views

Solving an inseparable initial value problem

Solve $t$$\frac{dx}{dt}$ = $x$ + $\sqrt{t^2 +x^2}$ with $x$(1) = 0. I tried to use substitution of $t = ux$ and ended up getting down to ( $\frac{1}{u\sqrt{u^2+1}}$ + $\frac{1}{u}$ ) $du$ = -$\frac{1}...
2
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1answer
100 views

How to derive the substituted Partial Differential Equation

I have a second order partial differential equation. $\frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial z^2} + \frac{2}{l}\frac{\partial U}{\partial z}=0$. I need to introduce a ...
1
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1answer
50 views

How isolate y from $a=\sqrt{(y^2+x^2)^3} + \sqrt{(y^2-x^2)^3}$

$$a=\sqrt{(y^2+x^2)^3} + \sqrt{(y^2-x^2)^3}$$ I need to isolate y in a family of such expressions. Can someone give me a guide to learn how deal with those... There's no further information or ...
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1answer
24 views

How to express this simple equation in terms of a different variable?

I am using the equation: $$y=(1-e^{-ax})*d$$ For the shape of a logarithmic attack phase of an audio synthesizer envelope. The idea is for it to go from (0,0) to (t,1) at a given time (x=t) value. ...
2
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1answer
68 views

$\int\sqrt{1-\tan x}~\mathrm{d}{x}.$ (Integral of a trigonometric function under square root)

$\int\sqrt{1-\tan x}~\mathrm{d}{x}.$ is an integral which I am not able to solve. I have restricted my ideas on trigonometric substitution but cannot conclude to an answer...will really appreciate if ...
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2answers
49 views

How Isolate y from $a=\sqrt{(y^2+(a+x)^2)^3}$ [closed]

Just that! I'm asking for a method to isolate $y$ from such expressions: $$a=\sqrt{(y^2+(a+x)^2)^3}$$ or even easier: $$a=\sqrt{(y^2+x^2)^3}.$$ EDIT: I'm just trying to revisit some of the basics ...
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1answer
19 views

Renaming integration variable in Fourier transformation

In my lecture on quantum field theory, we have recently discussed the canonical quantisation of a scalar field. There is one particular calculation, that I do not quite understand. Namely, we have ...
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1answer
20 views

Verification by direct substitution using matrices of ODE

I have this matrix: $$ \overline X=\begin{pmatrix} c_1e^{-2t} +2c_2e^{5t} \\ -3c_1e^{-2t}+c_2e^{5t} \\ \end{pmatrix} $$ I want to verify that the above is a soltuion to $$ \...
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1answer
101 views

Olympiad Inequality with Condition

I would like to prove this : Let $x,y,z$ be positive real numbers such that $xyz=1$ then we have : $$\frac{\left(\frac{1}{x}+\frac{z}{x}+z\right)\left(1+\frac{1}{x}+z\right)}{3\left(\frac{z}{x}\...
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3answers
66 views

Prove $\int f(x)f'(x)\,dx = \frac {[f(x)]^2}2 + c$ through substitution

I've always taken integration by substitution for granted but recently I've learned that differentials can't fully be treated as variables and that the process of integration by substitution is really ...
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5answers
53 views

Integration question (don't know what to substitute).

How do you integrate the following? $$\int\left(\frac{1+x}{1-x}\right)^{\frac{1}{2}}dx$$ Please do not give the full answer, but just what to start with. Thanks.
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2answers
56 views

Solve $\int_0^{\infty} x^2 \frac{x}{\theta}e^{\frac{x^2}{2\theta}} dx $

Solve $$\int_0^{\infty} x^2 \frac{x}{\theta}e^{\frac{x^2}{2\theta}} dx $$ Solution: $$\int_0^{\infty} x^2 \frac{x}{\theta}e^{\frac{x^2}{2\theta}} dx = - \int_0^\infty x^2 de^{\frac{-x^2}{2\theta}}dx^...
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0answers
52 views

Find the integral of the following function.

Question. My approach: Unfortunately I am stuck right after this step. I cannot determine what kind of "substitution" would bring the denominator as in the whole function in any solvable form like; ...
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1answer
28 views

How to use algebra to solve this?

Use Algebra to solve for $x$: $8(9^x)+3(6^x)-81(4^x)=0$ My attempt was to decompose the exponential functions into $2^x$ and $3^x$ where possible and then substitute them for some $p$ and $q$. I was ...
2
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4answers
84 views

Let $a, b, c \in \mathbb{R^+}$ and $abc=8$ Prove that $\frac {ab+4}{a+2} + \frac {bc+4}{b+2} + \frac {ca+4}{c+2} \ge 6$

Let $a, b, c \in \mathbb{R^+}$ and $abc=8$ Prove that $$\frac {ab+4}{a+2} + \frac {bc+4}{b+2} + \frac {ca+4}{c+2} \ge 6$$ I have attempted multiple times in this question and the only method that I ...
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2answers
63 views

Find maximum value

Given $0 \leq a,b,c \leq \dfrac{3}{2}$ satisfying $a+b+c=3$. Find the maximum value of $$N=a^3+b^3+c^3+4abc.$$ I think the equality does not occur when $a=b=c=1$ as usual. I get stuck in finding the ...
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2answers
86 views

Maximize the value of $\sqrt{x-x^2}+\sqrt{cx-x^2}$ without using calculus

Assume that $c$ is positive. How can we maximize the value of $\sqrt{x-x^2}+\sqrt{cx-x^2}$ with respect to $x$ without the use of calculus? With calculus, we can easily find out that the max of the ...
4
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2answers
63 views

Calculus u-substitution: Choosing between $\frac{dx}{du}$ and $\frac{du}{dx}$

I searched and couldn't find any answers to this one. It's probably a dumb question but one that has been troubling me. Let's say I have an integration: $$\int \frac{1}{3x+2}$$ It seems the correct ...
4
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1answer
55 views

Logarithmic equation $\log_2(x+4)=\log_{4x+16}8$

So the problem goes: What is the product of all solutions in the equation $$\log_2(x+4)=\log_{4x+16}8$$ The solution to this should be $31\over4$, but I got $-14$. This is what I did: \begin{...
2
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3answers
63 views

For u-substitution of a definite integral, is it acceptable to have the limits (bounds) of the new integral equal to one another?

For example, I have the problem: $$ \int_{-1}^1x^4\left(1-x^2\right)^2dx $$ I set $\,u=1-x^2\,$, which gives $\,x^2=1-u\,$, $\,x=\pm\sqrt{1-u}\,$ and $\,-\frac{1}{2}du=xdx$. For the upper and lower ...
2
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1answer
39 views

How do you substitute an arbitrary conic equation with trigonometric functions?

For example, Circle ($\frac{x^2}{a^2}+\frac{y^2}{a^2}=1$) : $$ \left\{ \begin{array}{c} x\to a\cos \theta \\ y\to a \sin \theta \\ \end{array} \right. $$ Oval ($\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$) : ...
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4answers
60 views

Integration by substitution vs. polynomial expansion first…different results??

I came across a textbook problem that showed an integral solved with the substitution method: $$\int_a^b(b-x)^2dx = \left(-\frac{(b-x)^3}{3}\right)\Biggl\vert_a^b$$ I then attempted to solve the ...
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1answer
34 views

Integration by substitution, finding a new formula in terms of the substiution

I am trying to solve a question that involves integration by substitution. Let $\displaystyle I = \int^{1}_{0}\frac{\sqrt{x}}{2- \sqrt{x}}dx$ Using the substitution $u = 2-\sqrt{x}$ show that $...
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1answer
60 views

How to prove the given inequality using CS or AM GM HM inequality [closed]

I was solving a problem on triangular inequalities and I have ended up with the following inequality required to be proven $$\sum_{cyc}\frac{x}{2x+y+z}\geq\frac{9\sqrt{3(x+y+z)xyz}}{4(x+y+z)^2},$$ ...
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0answers
39 views

Matrix Operations and Dependent Variables

Consider the following systems along with the corresponding substitutions: \begin{align} \text{System 1} \\ 2x^2 + 3y + 5z^z & = 7 \\ 4x^2 + 9y + 10z^z & = 2 \\ x^2 + y = 2z^z & = 1 \end{...
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3answers
137 views

Roots of $2x^3-4x+1$

I'm having difficulty getting the solution to the cubic equation $2x^3-4x+1=0$ and from http://www2.trinity.unimelb.edu.au/~rbroekst/MathX/Cubic%20Formula.pdf it claims that the general solution to $...
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1answer
24 views

Why can you substitute both the real and imaginary parts of an equation separately into another?

The title is very vague, apologies. I'm taking A Level Further Mathematics, and we've come to the end of our topic on Argand Diagrams. There is a question (which I'm not looking for the answer to), ...
4
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1answer
111 views

Prove: for all $x \in (0, 1], 2^x+2^{\frac{1}{x}} \leqslant 2^{x+\frac{1}{x}}$

This problem is from my math teacher.I tried using Calculus, the derivative function is like a black hole.Then I graphed it by Mathematica. As the following picture shows, I was strongly astonished. ...
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votes
3answers
43 views

Non-linear first order ODE with auxiliary variable

I have a problem with this equation: $y'(x)=\frac{2y(x)-x}{2x-y(x)}$. Using $y=xz$ i'm arrived to prove that $\frac{z-1}{(z+1)^{3}}=e^{2c}x^{2}$, but now i'm stuck. How can i explain the $z$? I've ...
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0answers
34 views

Replacing known “things” in differential equations

So I have the following differential equation: \begin{equation} \ddot x + \frac{b}{m}\dot x + \frac{2k}{m}x = 0 \end{equation} Let's say I also find that $a(t) = -\ddot x$, which helps me to reach ...
2
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1answer
39 views

Legitimate change of variable in integral of an integral function?

Consider the following expression: $$ A^q=\int_{\mathbb{R}^{n}}\left(\int_{\mathbb{R}^{n}}\left|f\left(x+y\right)g\left(x+y+Cy\right)\right|^{p}dy\right)^{q/p}dx, $$ where $1\leq p,q<\infty$, $C$ ...
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0answers
127 views

What is more historically correct?

I was wondering what was formally correct, when resolving an integral by substitution, use as a new variable $t$ or $u$? I noticed that here in Italy we use $t$, while I saw that abroad (e.g USA) use $...
3
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4answers
57 views

Non-linear second order ODE

I have to solve $$ y''(x)+(y'(x))^2=y'(x). $$ Using $ y'(x)=z $, I can write $$\int \frac{1}{z-z^2}dz=\int dx $$ So: $$\frac{1}{z(1-z)}=\frac{A}{z}+\frac{B}{1-z}$$ leads to $$ \int \frac{1}{z(1-z)...
2
votes
1answer
87 views

How to prove inequality $(a^2+b^2+c^2)^3\ge6(a^3+b^3+c^3)^2$ when $a+b+c=0$?

I know the identity $a^3+b^3+c^3-3abc = 1/2 (a+b+c) [(a-b)^2+(b-c)^2+(c-a)^2]$. So when it comes to this problem where $a+b+c=0$, you get $(a^2+b^2+c^2)^3\ge54(abc)^2$ when $a+b+c=0$ (because $ a^3+b^...
0
votes
1answer
32 views

unknown square root substitution in integration

The integral $$ \int d\theta=\int\frac{r_0}{R}\sqrt{\frac{R^2-r^2}{r^2-r_0^2}}\frac{dr}{r} \tag{1} $$ appears in the solution of Venezian to the terrestrial brachistochrone problem. (Venezian, ...
1
vote
3answers
51 views

Substitution in $\int \cos^3(x)$

I have an problem with the substitution in $$\int \cos^3(x)dx$$ Ive seen some answers where the integration is done by substituting $$\int \cos^3(x) = \int \cos(x)(1-\sin^{2}(x))$$ $u = \sin(x)$ ...
2
votes
1answer
61 views

Given triangle sides $a$, $b$, $c$, show $\sum_{\text{cyc}}\frac{a}{a+b}\geq\frac32\prod_{\text{cyc}}(\frac{a}{a+b}+\frac{b}{b+c})$

If $a,b,c$ are three sides of a triangle then I need to prove that $$ {\frac {a}{a+b}}+{\frac {b}{b+c}}+{\frac {c}{c+a}}\geq \frac{3}{2}\, \left( { \frac {a}{a+b}}+{\frac {b}{b+c}} \right) \left( {\...
2
votes
4answers
95 views

Prove that if in a tetrahedron if two pairs of opposite edges are perpendicular then the third pair is also perpendicular.

Prove that if in a tetrahedron if two pairs of opposite edges are perpendicular then the third pair is also perpendicular. Method: $\vec a+ \vec b + \vec c + \vec d +\vec e + \vec f = 0 \tag0$ ...
2
votes
1answer
74 views

Show that $\sum_{\text{cyc}} \frac{1}{b^2+c^2+5bc-a^2} \leq \frac{\sqrt3}{8S}$ for a triangle with sides $a$, $b$, $c$ and area $S$

Let be $a$, $b$, $c$ sides of a triangle and $S$ his area. Prove that $$\sum_{\text{cyc}} \frac{1}{b^2+c^2+5bc-a^2} \leq \frac{\sqrt3}{8S}$$ My idea: $b^2+c^2-a^2 = 2bc \cos A$, so the inequality is ...
0
votes
0answers
14 views

check of calculation of a surface integral

Find surface integral $$I= \int _M (x+y+z) \mathrm{d} S$$, where M is the upper half-sphere given explicitly as $z=\sqrt{a^2-x^2-y^2}$, $x^2+y^2 < a^2$. I would like you to check if my ...
0
votes
1answer
20 views

System of 2nd order diff equations with $x_1(t), x_2(t)$

Could you help me with the following system of two equations? I have browsed through the forum, but I haven't found a similar problem. $$m_1x_1''(t) + k(x_1 - x_2)+ cx_1'(t) = 0$$ $$m_1x_2''(t) + k(...