What is Substitution: Definition and 815 Discussions

A substitution reaction (also known as single displacement reaction or single substitution reaction) is a chemical reaction during which one functional group in a chemical compound is replaced by another functional group. Substitution reactions are of prime importance in organic chemistry. Substitution reactions in organic chemistry are classified either as electrophilic or nucleophilic depending upon the reagent involved, whether a reactive intermediate involved in the reaction is a carbocation, a carbanion or a free radical, and whether the substrate is aliphatic or aromatic. Detailed understanding of a reaction type helps to predict the product outcome in a reaction. It also is helpful for optimizing a reaction with regard to variables such as temperature and choice of solvent.
A good example of a substitution reaction is halogenation. When chlorine gas (Cl2) is irradiated, some of the molecules are split into two chlorine radicals (Cl•) whose free electrons are strongly nucleophilic. One of them breaks a C–H covalent bond in CH4 and grabs the hydrogen atom to form the electrically neutral HCl. The other radical reforms a covalent bond with the CH3• to form CH3Cl (methyl chloride).

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  1. A

    Trigonometric substitution, What am i doing wrong?

    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...
  2. J

    I Solving a system of linear equations using back substitution

    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...
  3. H

    A How to get Peierls substitution in edge state?

    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?
  4. DeathbyGreen

    A Pierels substitution integral approximation

    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...
  5. Q

    Difficulties with Substitution Rule (integration)

    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...
  6. grquanti

    I Substitution in partial differential equation

    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...
  7. SamRoss

    B Not following an integral solution

    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?
  8. H

    B Does integration commute with substitution t=0?

    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...
  9. uchuu-man chi

    I Need a little push on this integral using trig substitution.

    ∫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...
  10. doktorwho

    Integral that is reduced to a rational function integral

    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...
  11. Conductivity

    B Integration by Substitution Using Infinite Sums

    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...
  12. doktorwho

    Integration by substitution question

    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...
  13. F

    MHB What are the next steps after finding a pattern in iterative substitution?

    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?
  14. M

    Calculus of Variations: interesting substitution

    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...
  15. chwala

    Integration problem using substitution

    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...
  16. J

    Trig substitution integration

    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...
  17. kolleamm

    B Derivatives: Solving a Substitution Error on MathsIsFun

    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!
  18. Rectifier

    Primitive function - smart substitution

    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?
  19. Rectifier

    Integration with variable substitution

    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}}...
  20. A

    MHB 2 Differential Equations by Substitution

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

    MHB Trig substitution question

    $$\int_{}^{} \frac{1}{x\sqrt{x^2 + 16}} \,dx$$ 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...
  22. A

    Substitution Rule for Integrals: How to Simplify Complex Integrands

    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...
  23. M

    Where am I going wrong in my radial equation substitution derivation?

    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) =...
  24. karush

    MHB W.8.7.23 int trig u substitution

    $\tiny{Whitman \ 8.7.23}$ \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]...
  25. S

    Question about the viability of a chemical reaction

    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...
  26. T

    MHB Integral using trig substitution

    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...
  27. B

    Trig Substitution Problem w/ tan substitution

    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.
  28. B

    Trigonometric Substitution Problem w/ Sin Substitution

    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))) /...
  29. Battlemage!

    Using substitution to turn a series into a power series.

    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...
  30. Battlemage!

    Integral of dn/(n^2 - 4) using trig substitution with sine

    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...
  31. karush

    MHB 206.8.8.49 trig substitution

    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...
  32. J

    Substitution reaction between methane and bromine

    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...
  33. N

    I Why use the Legendre transform over back substitution

    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...
  34. karush

    MHB UHWO 242 integral substitution

    $\large{S6.7.r.44}$ $$\displaystyle I=\int_{2}^{6}\frac{y}{\sqrt{y-2}} \,dy = \frac{40}{3}$$ $$ \begin{align} u&=y-2 &y&=u+2 \\ du&=dy \end{align}$$ 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...
  35. DavideGenoa

    I Substitution in a Lebesgue integral

    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...
  36. karush

    MHB S6-7.1.79 log integral u substitution

    $\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🐮...
  37. T

    MHB Integral using trig substitution

    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} =...
  38. T

    MHB Solving an integral with trigonometric substitution

    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?
  39. karush

    MHB LCC 206 {review 7r5} Integral 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 } $...
  40. karush

    MHB LCC 206 {review 7r13} Integral substitution

    $\tiny\text{LCC 206 {review 7r13} Integral substitution}$ $$I=\int_{0}^{\pi/4} \cos^5\left({2x}\right)\sin^2 \left({2x}\right)\,dx= \frac{4}{105} $$ $\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}...
  41. karush

    MHB Steward e6 {7r11} Integral substitution

    $\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...
  42. karush

    MHB LCC 205 8.4.12 sine substitution

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

    MHB Can Trig Substitution be used to Solve Trigonometric Integrals?

    {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...
  44. karush

    MHB LCC 205 8.4.7 sine substitution (definite integral)

    $\tiny{LCC \ \ 205 \ \ \ 8.4.7 \ \ sine \ \ substitution }$ $\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...
  45. karush

    MHB Trig Substitution for Whitman 8.4.8: Complete the Square

    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...
  46. N

    Integrating with Substitution: Solving for ∫(3x^2+x)(2x^3+x^2)^2 dx

    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...
  47. RoboNerd

    I Question on basic trig substitution with x = sin theta

    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...
  48. Alexandra Fabiello

    Concept question - integrals requiring substitution to solve

    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...
  49. Edison Bias

    I Integral problems with polar coordinates and variable substitution

    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
  50. G

    What substitution to use in this type of integral?

    Homework Statement Find the integral \int \frac{3x+1}{(x^2-x-6)\sqrt{3x^2+4x-7}}\mathrm dx 2. The attempt at a solution I have tried the types of substitutions of irrational functions, and Euler substitutions. However, it seems that nothing simplifies this integral. What substitution is...
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