Substitution Definition and 816 Threads

This is a small part of a question from the book, so I think the format does not really apply here. When doing questions for solving differential equation with substitution, I encountered a substitution ## y(x)=\frac{1}{v(x)} ##. And I am not sure about the calculus in finding ## \frac{dy}{dx}...

Homework Statement Homework Equations The Attempt at a Solution Here is my answer, i get 1/24 For my first step i divided both terms under the radical by 4, then split 1/4 into (1/2)2, i saw something very similar in my book so i did the same thing, but i just realized this has to be...

Hello, its been a while since I have taken linear algebra and I am having trouble understanding what a target vector is. I need to solve a system of linear equations in matrix form using back substitution and with different target vectors. I don't have a problem with back substitution, but I...

In paper PRL 101, 246807 (2008), authors use "Peierls substitution", that is ky -> -i∂ y. As we know, ky is eigenvalue of translation operator in period potential, while -i∂ y is momentum operator, it seems they are huge different. So I wonder how to get ""Peierls substitution" in strict math way?

In the textbook "Topological Insulators and Topological Superconductors" by B. Andrei Bernevig and Taylor L. Hughes, there is a chapter titled "Hall conductance and Chern Numbers". In section 3.1.2 (page 17) they are discussing including an external field in a tight binding model, the Peierls...

I can obviously do the chain rule and see how the final expression of the derivative is related to the original function but I can't seem to figure out the substitution Rule as an intuitive way of solving the indefinite integral of functions... bear with me if I'm too verbose, I've attached an...

Hello everybody. Consider $$\frac{\partial}{\partial t}f(x) + ax\frac{\partial }{\partial x}f(x) = b x^2\frac{\partial^2}{\partial x^2}f(x)$$ This is the equation (19) of...

In the image below, why is the third line not \frac {ln(cosx)} {sinx}+c ? Wouldn't dividing by sinx be necessary to cancel out the extra -sinx that you get when taking the derivative of ln(cosx)? Also, wouldn't the negatives cancel?

Let ##g(x,t)=\int f(k,x,t)\,dk## Under what conditions is the following true? ##g(x,0)=\int f(k,x,0)\,dk## That is, we can get the value of ##g(x,t)## when ##t=0##, by (1) either substituting ##t=0## into ##g(x,t)## or (2) by first substituting ##t=0## into ##f(k,x,t)## and then integrating...

∫x2√(3+2x-x2) dx Here's what I've already done: completed the square ∫x2√(4-(x-1)2) dx (x-1) = 2sinθ sinθ = (x-1)/2 x = 2sinθ+1 dx = 2cosθ dθ trig sub + pulled out constants 4∫(2sinθ+1)2√(1-sin2θ)cosθ dθ trig identity 4∫(2sinθ+1)2√(cos2θ)cosθ dθ 4∫(2sinθ+1)2(cos2θ)dθ expanded + trig...

Homework Statement Suggest an integral that is reduced to a rational function integral when this substitution is used: ##a)## ##t=\sin x## ##b)## ##t=\sqrt[6] {x+5}## ##c## ##\sqrt{1-9x^2}=-1+xt## Homework Equations 3. The Attempt at a Solution [/B] I found this to be a very interesting...

I have seen the wikipedia's proof which can be found here: https://proofwiki.org/wiki/Integration_by_Substitution However sometimes, we have problems where you have a ##d(x)## times ## f(g(x))## times g prime of x where we use substitution and it works but the proof didn't prove this...

Homework Statement Question: To solve the integral ##\int \frac{1}{\sqrt{x^2-4}} \,dx## on an interval ##I=(2,+\infty)##, can we use the substitution ##x=\operatorname {arcsint}##? Explain Homework Equations 3. The Attempt at a Solution [/B] This is my reasoning, the function ##\operatorname...

Hi i Have this equation: T(n)=2T(n/2)+n^2 I understand for iterative substitution you need to find patterns so here's what i got: 2^2T(n/2^2)+n2/2+n^2 2^3T(n/2^3)+n2/2^2+n2/2+n^2 My question is what to do after you have found the pattern?

Homework Statement Find the externals of the functional $$\int\sqrt{x^2+y^2}\sqrt{1+y'^2}\,dx$$ Hint: use polar coordinates. Homework Equations ##x=r\cos\theta## ##y=r\sin\theta## The Attempt at a Solution Transforming the given functional where ##r=r(\theta)## yields...

Homework Statement using ## u= sin 4x## find the exact value of ##∫ (cos^3 4x) dx##[/B]Homework EquationsThe Attempt at a Solution ## u= sin 4x## [/B]on integration ##u^2/2=-cos4x/4 ## , →##-2u^6={cos 4x}^3 ##...am i on the right track because now i end up with...

Homework Statement ∫8cos^3(2θ)sin(2θ)dθ Homework EquationsThe Attempt at a Solution rewrote the integral as: 8∫(1-sin^2(2θ))sin(2θ)cos(2θ)dθ u substitution with u=sin(2θ) du=2cos(2θ)dθ 4∫(1-u^2)u du= 4∫u-u^3 du 4(u^2/2-u^4/4)+C undo substitution and simplify 2sin^2(2θ)-sin^4(2θ)+C The book...

I'm learning about derivatives and on the website they put the value x^2 into f(x + dx) and it became (x + dx)^2 Shouldn't it be (x^2 + dx^2) ? It's the last example https://www.mathsisfun.com/calculus/derivatives-dy-dx.html Thanks in advance!

The problem $$ \int \frac{x}{\sqrt{x^2+2x+10}} \ dx $$ The attempt ## \int \frac{x}{\sqrt{x^2+2x+10}} \ dx = \int \frac{x}{\sqrt{(x+1)^2+9}} \ dx## Is there any smart substitution I can make here to make this a bit easier to solve?

Hello, I am having trouble with solving the problem below The problem Find all primitive functions to ## f(x) = \frac{1}{\sqrt{a+x^2}} ##. (Translated to English) The attempt I am starting with substituting ## t= \sqrt{a+x^2} \Rightarrow x = \sqrt{t^2 - a} ## in $$ \int \frac{1}{\sqrt{a+x^2}}...

solve the following differential equation with the suggested change of variables.

$$\int_{}^{} \frac{1}{x\sqrt{x^2 + 16}} \,dx$$ I can set $x = 4 tan\theta$. Thus $dx = 4 sec^2 \theta d\theta$ So, plug this into the first equation: $$\int_{}^{} \frac{4 sec^2 \theta }{4 tan\theta \sqrt{16 tan^2\theta + 16}} \,d\theta$$ Then, $$\int_{}^{} \frac{ sec^2 \theta }{4 tan\theta...

Homework Statement $$\int_{0}^{2} r\sqrt{5-\sqrt{4-r^2}} dr$$ Homework EquationsThe Attempt at a Solution would i substitute ##u=4-r^2##? After of which I would input into the integral and get: $$\int_{0}^{2} \sqrt{5-\sqrt{u}}du$$ What would I do here? Do I just work inside the radical(so 5r...

Homework Statement Essentially we are describing the ODE for the radial function in quantum mechanics and in the derivation a substitution of u(r) = rR(r) is made, the problem then asks you to show that {(1/r^2)(d/dr(r^2(dR/dr))) = 1/r(d^(2)u/dr^2) Homework Equations The substitution: u(r) =...

$\tiny{Whitman \ 8.7.23}$ \begin{align} \displaystyle I&=\int \sin^3(t) \cos^2(t) \ d{t} \\ u&=\cos(t) \therefore du=-\sin(t) \, dt \\ \textit{substitute $\cos(t)=u$}&\\ I_u&=-\int (1-u^2) u^2 \, du=-\int(u^2-u^4) \, dt\\ \textit{integrate}&\\ &=-\left[\frac{u^3}{3}-\frac{u^5}{5}\right]...

For organic chemistry we typically look at the stability of the end product to determine if the reaction will proceed. For instance for an Sn2 reaction we would check if the product anion is more stable or less stable than the nucleophile attacking - if the product anion is more stable than the...

I have $$\int_{}^{} \frac{1}{\sqrt{1 - x^2}} \,dx$$ I can let $x = \sin\left({\theta}\right)$ then $dx = cos(\theta) d\theta$ then: $$\int_{}^{} \frac{cos(\theta) d\theta}{\sqrt{1 - (\sin\left({\theta}\right))^2}}$$ Using the trig identity $1 - sin^2\theta = cos^2\theta$, I can simplify...

Homework Statement Under #3 Homework Equations Trig identities The Attempt at a Solution The picture attached is my attempt. The square in the upper upper left is the problem and the one in the lower right is my solution. I'm seeing that I'm getting the wrong answer, but not how.

Homework Statement ∫(√(64 - x^2)) / x dx I must solve this using a sin substitution. Homework Equations x = 8sinΘ dx = 8cosΘ dΘ Θ = arcsin(x/8) Pythagorean Identities The Attempt at a Solution (After substitution) = ∫8cosΘ * (√(64 - 64sin^2Θ)) / 8sinΘ dΘ = ∫(cosΘ * (√(64(1 - sin^2Θ))) /...

Homework Statement The problem asks to use a substitution y(x) to turn a series dependent on a real number x into a power series and then find the interval of convergence. \sum_{n=0}^\infty ( \sqrt{x^2+1})^n \frac{2^n }{3^n + n^3} Homework Equations After making a substitution, the book...

Homework Statement \int_{}^{∞} \frac{1}{n^2 - 4} dn Homework Equations I'm trying to do this a way that it isn't usually done. Normally this is done with partial fractions. I'm trying to do it by using trig substitution using sine, which requires some algebraic manipulation. For some reason...

206.8.8.49 $a=0 \ \ b=12$ $\displaystyle I_{49}=\int_{a}^{b} \frac{dx}{\sqrt[]{144-x^2}} \, dx = \arcsin{\left[\frac{1}{12}\right]} \\$ $ \text{use identity} $ $\sin^2\theta+\cos^2\theta = 1 \Rightarrow 1-\cos^2\theta=\sin^2\theta \\$ $\text{x substituion} $ $\displaystyle...

We can react one mole of ##CH_4## and one mole of ##Br_2## to obtain the following equation: ## CH_4 + Br_2 \rightarrow CH_3 Br + HBr ## A single bromine atom switches places with a single hydrogen atom. Now, if we supply a greater amount of ##Br_2## while keeping the amount of ##CH_4## the...

In thermodynamics we use a variation of the Legandere Legendre transform to move from one description of the system to another ( depending on what is the control variable...), but I don't understand why choose to use the Legandere Legendre transform over writing x in terms of s=dy/dx and back...

$\large{S6.7.r.44}$ $$\displaystyle I=\int_{2}^{6}\frac{y}{\sqrt{y-2}} \,dy = \frac{40}{3}$$ $$ \begin{align} u&=y-2 &y&=u+2 \\ du&=dy \end{align}$$ then $$\displaystyle I=\int_{0}^{4}\frac{u+2}{\sqrt{u}} \, du =\int_{0}^{4}{u}^{1/2} \, du + 2\int_{0}^{4} {u}^{-1/2} \, du$$ Just seeing if...

Hi, friends! I read that, if ##f\in L^1[c,d]## is a Lebesgue summable function on ##[a,b]## and ##g:[a,b]\to[c,d]## is a differomorphism (would it be enough for ##g## to be invertible and such that ##g\in C^1[a,b]## and ##g^{-1}\in C^1[a,b]##, then...

$\large{7.R.79} $ $\tiny\text{UHW 242 log integral }$ https://www.physicsforums.com/attachments/5717 $$\begin{align} \displaystyle x& = \frac{1}{u} & {u}^{2 }du&={dx } \end{align} $$ $$I=\int_{0}^{\infty} \frac {\ln\left({\frac{1}{u}}\right)} {1+\frac{1}{{u}^{2 }}} {u}^{2} \,du \\ Stuck🐮...

I have this integral: $$\int_{}^{}\frac{1}{x^2 - 9} \,dx$$ I believe I can use trig substitution with this so I can set $x = 3 sec\theta$ Evaluating this, I get $$ln|\csc\left({\theta}\right) - \cot\left({\theta}\right)| + C$$ Since $x^2 - 9 = 9sec^2\theta - 9$, then $\frac{x^2 - 9}{3} =...

I have this integral: $$\int_{}^{} \frac {x^2}{{(4 - x^2)}^{3/2}}\,dx$$ I can see that we can substitute $x = 2sin\theta$, and $dx = 2cos\theta d\theta$, but I am unable to see how $\sqrt{4 - x^2} = 2cos\theta$. How can I get this substitution?

$\tiny\text{LCC 206 {review 7r5} Integral substitution}$ $$I=\int_{-2} ^{3} \frac{3}{{x}^{2}+4x+13}\,dx= \arctan {\left(\frac{5}{3}\right)} $$ $\text{ complete the square } $ $$ \frac{3}{{x}^{2}+4x+13} \implies\frac{3}{{\left(x+2 \right)}^{2}+{3}^{2}}$$ $\text{ then } $...

$\tiny\text{LCC 206 {review 7r13} Integral substitution}$ $$I=\int_{0}^{\pi/4} \cos^5\left({2x}\right)\sin^2 \left({2x}\right)\,dx= \frac{4}{105} $$ $\text{u substitution } $ \begin{align}\displaystyle u& = 2x& du&= 2 \ d{x}& 2(\pi/4) &=\pi/2 \end{align} $\text{then } $ $$I=2\int_{0}^{\pi/2}...

$\tiny\text{Steward e6 {7r11} } $ $$\displaystyle I=\int_1^2 {\frac{\sqrt{{x}^{2}-1}}{x} } \ d{x} = {\sqrt{3}+\frac{1}{3}\pi} $$ Again what substitution was the question but tried.. $$\begin{align} u& = {x}^{2}-1 & du&= 2x \ d{t} & x&=\sqrt{u+1} \end{align}$$ $$\displaystyle I=\int_1^2...

$\tiny{LCC \ \ 205 \ \ \ 8.4.12 \ \ sine \ \ substitution }$ $$\displaystyle I=\int_{1/2}^{1} \frac{\sqrt{1-{x}^{2 }}}{{x}^{2 }} \ dx $$ Substitution $$\displaystyle x = \sin \left({u}\right) \ \ \ dx=\cos\left({u}\right)\ du $$ Change of variables $$\arcsin\left({1/2...

{8.7.4 whit} nmh{962} $$\displaystyle I=\int \sin\left({t}\right) \cos\left({2t}\right) \ dt $$ substitution $u=\cos\left({t}\right) \ \ \ du=-\sin\left({t}\right) \ dt \ \ \ \cos\left({2t}\right) =2\cos^2 \left({t}\right)-1 $ $\displaystyle I=\int \left(1-2u^2 \right) du \implies...

$\tiny{LCC \ \ 205 \ \ \ 8.4.7 \ \ sine \ \ substitution }$ $\displaystyle I=\int_0^{5/2} \frac{1}{\sqrt{25-x^2}}\ dx =\frac{\pi}{6} $ $$\displaystyle x=5\sin\left({u}\right) \ \ \ dx=5\cos\left({u}\right) \ \ du $$ $$\displaystyle I=\int_0^{5/2} \frac{1}{\sqrt{25-25 \sin^2...

Whitman 8.4.8 Trig substitution? Whitman 8.4.8 Complete the square.. \begin{align*} \int\sqrt{x^{2}-2x}dx &=\int\sqrt{x^{2}-2x+1-1}dx\\ &=\int\sqrt{(x-1)^{2}-1^{2}}dx\\ &=\int\sqrt{U^{2}-1^{2}}dx\\ \end{align*} Was wondering what substation best to use...

Homework Statement dz/dx=(3x2+x)(2x^3+x^2)^2[/B]Homework Equations ∫(3x^2+x)(2x^3+x^2)^2 dx The Attempt at a Solution I tried substituting (2x^3+x^2) Let t= 2x^3 + x^2 dt=6x^2+2x dx dt/dx= 6x^2+2x I can only solve till this point . I don't have any clue how to solve it further But how do we...

Say I have the integral of [ 1 / ( sqrt( 1 - x^2) ] * dx . Now I was told by many people in videos that I substitute x = sin theta, and this has me confused. Wouldn't I need to substitute x = cos theta instead? as x = cos theta on the unit circle instead of sin theta? Thanks in advance for...

Homework Statement Random example: 2[([x^2] + 3)^7](7x) Homework Equations ? The Attempt at a Solution I know that somehow you substitute [x^2] + 3 with 'u', but I don't understand the process going forward for it, and my teacher and textbook has some rather convoluted stuff in it, so if...

H! I wonder how to solve: I=\int_{-\infty}^{\infty}e^{-u^2}\frac{1}{1+Cu} du I have solved: \int_{-\infty}^{\infty}e^{-u^2}du which equals \sqrt{\pi} and I solved it with polar coordinates and variable substitution. Thankful for help! Edison