Substitution Definition and 816 Threads

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. M

    I Question about this Integration by Substitution

    This is part of the working from f(3x^2-1)^2xdx; I don't understand from when 6x becomes 1/6
  2. chwala

    Use substitution to solve the definite integral

    I have ##1-x^2 = 1- \sin^2 θ = \cos^2 θ## and ## dx =cos θ dθ## ##\int_0^{0.5} (1-x^2)^{1.5} dx = \int_0^{\frac{π}{6}} [cos ^2θ]^\frac{3}{2} dθ = \int_0^{\frac{π}{6}} [cos ^4θ] dθ## Suggestions on next step.
  3. deuteron

    Potential of a rotationally symmetric charge distribution

    First, we rewrite the term ##|\vec r-\vec r_q|## in the following way: $$|\vec r-\vec r_q|= \sqrt{(\vec r-\vec r_q)^2} = \sqrt{\vec r^2 + \vec r_q^2 -2\vec r\cdot\vec r_q} = \sqrt{r^2 + r_q^2 -2rr_q\cos\theta}$$ Due to rotational symmetry, we go to spherical coordinates: $$\phi_{e;\vec r_q} =...
  4. murshid_islam

    I What's my mistake in this integration problem?

    Here's the problem: ##\int_0^{2\pi} \cos^{-1}(\sin(x)) \mathrm{d}x## If I do the substitution ##u = \sin(x)##, both the limits of integration become 0 and the integral would result in 0. But the graph of the function tells me the area isn't 0. Where am I going wrong?
  5. Safinaz

    How to Approach Solving a Nonlinear Second Order ODE with a Quadratic Term?

    I know how to solve similar ODEs like ## \frac{\partial^2 x}{ \partial t^2} + b \frac{\partial x}{ \partial t} + C x =0 ## Where one can let ## x = e^{rt}##, and the equation becomes ## r^2 + b r + C =0 ## Which can be solved as a quadratic equation. But now the problem is that there is...
  6. Ducatidragon916

    Research on HV to UHV vacuum gauge and controller build (Amateur scale)

    How did you find PF?: Looking for Circuits on Pirani Gauges I am in the process of building a High Vacuum system and obtained an outdated Pirani Gauge controller TM120 on ebay. The unit is from Leybold Heraeus company and is built with quality. I contacted them on information on the manuals...
  7. chwala

    Is the method used to evaluate the given integral correct?

    Method 1, Pretty straightforward, $$\int_{-1}^0 |4t+2| dt$$ Let ##u=4t+2## ##du=4 dt## on substitution, $$\frac{1}{4}\int_{-2}^2 |u| du=\frac{1}{4}\int_{-2}^0 (-u) du+\frac{1}{4}\int_{0}^2 u du=\frac{1}{4}[2+2]=1$$ Now on method 2, $$\int_{-1}^0 |4t+2| dt=\int_{-1}^{-0.5} |4t+2|...
  8. 1

    Integration Substitution Techniques for quadratic expressions under square roots

    Hi, With respect to the techniques mentioned in point 2 and 3: Can someone explain or even better, post a link for an explanation or a videos showing the use of these two techniques. Below excerpt shows problems 4 and 5 referenced in the above 2 points:
  9. A

    I Where did this substitution technique go wrong?

    We can solve ##y'(x) = (ax+b)y(x)## by rearranging to obtain ##\frac{y'}{y} = ax +b## and solving in terms of an exponential. I tried an alternative technique to see if it would work, and somewhere I went wrong. The point of the technique is that a slightly simpler version of the problem should...
  10. R

    B Are Both Answers Correct for Trigonometric Substitution Integral?

    Last night I tried to calculate from an automatically generated Wolfram Alpha problem set: $$\int{\frac{1}{\sqrt{x^2+4}}}dx$$ I answered $$\ln({\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}})+C$$ The answer sheet gave: $$\ln({\sqrt{x^2+4}+x})+C$$ I couldn't see where I had gone wrong, so I tried...
  11. J

    I Confusion about the Substitution rule

    Given is a function ##P(E)## and its derivative ##f(E)##. Writing ##E## in terms of ##v## according to ##E=\frac{1}{2}mv^2## gives the derivative ##g(v)=f(E)mv## and ##dE=mvdv##. My issue arises from the premise that I learned; Integrals and derivatives are based on steps of a fixed constant...
  12. karush

    Solving this integral with u substitution

    Evaluate ##\displaystyle\int_{0}^{3}\frac{x+3}{\sqrt{x^{3}+1}}dx+5## W|A returned 11.7101 ok subst is probably just one way to solve this so ##u=x^{3}+1 \quad du= 3x^2##
  13. B

    MHB Antidifferentiation by Substitution

    1.\[ \int x^2 e^{x^3} dx \] 2. \[ \int sin(2x-3)dx \] 3. \[ \int (\cfrac {3dx}{(x+2)\sqrt {x^2+4x+3}} ) \] 4. \[ \int (\cfrac {x^3}{(x^2 +4)^\cfrac {3}{2}} )dx \]
  14. S

    Integral using substitution x = -u

    Is it possible to solve this integral? I think the substitution ##x=-u## does not help at all since it only changes variable ##x## to ##u## without changing the integrand much. Using that substitution: $$\int \frac{6x^2+5}{1+2^x}dx=-\int \frac{6u^2+5}{1+2^{-u}}du$$ Then how to continue? Thanks
  15. Einstein44

    Line of regression substitution

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  16. A

    Calculus Textbook for Integration using Hyperbolic substitution

    Can someone please tell me the book that contain integration using hyperbolic substitution for beginner? I know that hyperbolic functions is taught in Calculus book but most of them is only some identities and inverses of hyperbolic functions.
  17. A

    I Integration Using Hyperbolic Substitution

    Can someone please show me an example of integration using hyperbolic substitution? Thank you
  18. J

    Show that the Diffusion Eqn after substitution gives the Helmholtz Eqn

    Question: Helmholtz as defined in text: My attempt so far
  19. Mayhem

    I Using v substitution for first order homogenous DE and constraining solution

    My considers a type of differential equation $$\frac{\mathrm{d} y}{\mathrm{d} x} = f\left(\frac{y}{x} \right )$$ and proposes that it can be solved by letting ##v(x) = \frac{y}{x}## which is equivalent to ##y = xv(x)##. Then it says $$\frac{\mathrm{d} y}{\mathrm{d} x} = v + x\frac{\mathrm{d}...
  20. JD_PM

    Minimal substitution on the Lagrangian of the complex KG field

    a) I think I got this one right. Please let me know otherwise We have (let's leave the ##x## dependence of the fields implicit :wink:) $$\mathscr{L} = N \Big(\partial_{\alpha} \phi \partial^{\alpha} \phi^{\dagger} - \mu^2 \phi \phi^{\dagger} \Big) = \partial_{\alpha} \phi^{\dagger}...
  21. hackedagainanda

    Simultaneous equations substitution method

    I'm really stuck on this one, I was able to get the answer but not by the substitution method. So its the weight as A and B so I get A + B = 24 A(3) = B(5) so in my head I calculate a few pairs, 3 x 5 = 15 but 3 + 5 only = 8 so the next pair would be 10 and 6 which is still to small so I move...
  22. cianfa72

    Circuit Theory - about the applicability of the substitution theorem

    Hi, I've a doubt about the applicability of the substitution theorem in circuit theory. Consider the following picture (sorry for the Italian inside it :frown: ) As far I can understand the substitution theorem can be applied to a given one-port element attached to a port (a port consists of...
  23. Addez123

    Solve this partial diff. equation using substitution

    I completely forgot how to solve these so here's my attempt: $$z = au + bv$$ $$z = a(x^2 + y^2) + be ^{-x^2/2}$$ $$z'_x = 2ax - bxe ^{-x^2/2}$$ $$z'_y = 2ay$$ Put that into the original equation and you get $$y * (2ax - bxe ^{-x^2/2}) -x * (2ay) = $$ $$-ybe^{-x^2/2} = xyz$$ $$z =...
  24. karush

    MHB 4.1.1 AP calculus Exam Int with U substitution

    Evaluate $\displaystyle\int{\dfrac{{(1-\ln{t})}^2}{t} dt=}$ $a\quad {-\dfrac{1}{3}{(1-\ln{t})}^3+C} \\$ $b\quad {\ln{t}-2\ln{t^2} +\ln{t^3} +C} \\$ $c\quad {-2(1-\ln{t})+C} \\$ $d\quad {\ln{t}-\ln{t^2}+\dfrac{(\ln{t^3})}{3}+C} \\$ $e\quad {-\dfrac{(1-\ln{t^3})}{3}+C}$ ok we can either expand...
  25. E

    Applying a substitution to a PDE

    Problem: Consider the equation $$\frac{\partial v}{\partial t} = \frac{\partial^{2} v}{\partial x^2} + \frac{2v}{t+1}$$ where ##v(x,t)## is defined on ##0 \leq x \leq \pi## and is subject to the boundary conditions ##v(0,t) = 0##, ##v(\pi, t) = f(t)##, ##v(x,0) = h(x)## for some functions...
  26. bagasme

    Wheatstone Bridge: Substitution Resistance Formula Derivation?

    Hello, In high school, I had been taught about finding substitution resistance from Wheatstone bridge. The formula: a. If the cross product of ##R1## and ##R3## is same as ##R2## and ##R4##, the galvanometer in the middle (##R_5##) can be omitted and use series-parallel principle to solve for...
  27. archaic

    B ##\int_a^b x^2\sin(2x)dx## by substitution

    Would this be valid manipulation for ##x\in[0,\,\pi/2]##? I know that it is integrable by parts, I just want to know where did the manipulation become invalid, if it did, and why. Thank you! $$\begin{align*} \mathrm I&=\int_a^b x^2\sin2x\,dx\\ &\text{I know that...
  28. jisbon

    Understanding Integration by Substitution

    Not sure how do I start from here, but do I let $$u = lnx$$ and substitute? Cheers
  29. karush

    MHB 2.6.62 inverse integrals with substitution

    ok this is from my overleaf doc so too many custorm macros to just paste in code but I think its ok,,, not sure about all details. appreciate comments... I got ? somewhat on b and x and u being used in the right places
  30. karush

    MHB 4.2.236 AP calculus Exam integral with u substitution

    AP Calculas Exam Problem$\textsf{Using $\displaystyle u=\sqrt{x}, \quad \int_1^4\dfrac{e^{\sqrt{x}}}{\sqrt{x}}\, dx$ is equal to which of the following}$ $$ (A)2\int_1^{16} e^u \, du\quad (B)2\int_1^{4} e^u \, du\quad (C) 2\int_1^{2} e^u \, du\quad (D) \dfrac{1}{2}\int_1^{2} e^u \, du\quad...
  31. chwala

    Reverse substitution to find the inverse of modular arithmetic

    ##132,289≡1973* 67 + 98## ##1973≡98*20+13## ##98≡13*7+7## ##13≡7*1+6## ##7≡6*1+1## now in reverse my attempt is as follows, ##1≡7-6## ## 1≡7-(13-7)## ##1≡2*7-(1973-20*132,289+1340*1973)## ##1≡2*7-(1341*1973-20*132,289## which is correct but my interest is in finding the inverse of 1973 help?
  32. Adgorn

    I Attempting to find an intuitive proof of the substitution formula

    Hello everyone. First off, I'm sorry if this post is excessively long, but after tackling this for so many hours I've decided I could use some help, and I need to show everything I did to express exactly what I wish to do. Also, to be clear, this post deals with integration by substitution. Now...
  33. F

    B How to Properly Use Substitution

    When substitution is properly used for a set of equations, I believe you get a new equation with solutions that are also solutions of both of the previous equations. The following equation has solutions x = 0 and x = 1. ##x=x^2## This next equation has solutions x = -2 and x = 2. ##x^2=4##...
  34. I

    Use a variable substitution to get into a Bessel equation form?

    Hello, For my homework I am supposed to get- into the form of a Bessel equation using variable substitution. I am just not sure what substitution to use. Thanks in advance.
  35. M

    An integration problem using trigonometric substitution

    This is the integral I try to take. ##\int\sqrt{1+9y^2}## and ##9y^2=tan^2\theta## so the integral becomes ##\int\sqrt{1+tan^2\theta}=\sqrt {sec^2\theta}##. Now I willl calculate dy. ## tan\theta=3y ## and ##y=\frac {tan\theta}3## and ##dy=\frac{1+tan^2\theta}3## Here is where I can only...
  36. echomochi

    Finding an implicit solution to this differential equation

    Homework Statement Find an equation that defines IMPLICITLY the parameterized family of solutions y(x) of the differential equation: 5xy dy/dx = x2 + y2 Homework Equations y=ux dy/dx = u+xdu/dx C as a constant of integration The Attempt at a Solution I saw a similar D.E. solved using the y=ux...
  37. Akash47

    Are all substitution reactions reversible?

    Consider a reaction: H2+CuCl2= Cu+2HCl This is a substitution reaction.But is this may not be a reversible reaction since Cu is less active than .So Cu can't substitute H from HCl and make a backward reaction.Is my thinking right?
  38. H

    Using hyperbolic substitution to solve an integral

    Homework Statement Homework Equations So the question is asking to solve an integral and to use the answer of that integral to find an additional integral. With part a, I don't have much problem, but then I don't know how to apply the answer from it to part b. I know I should subsitute all...
  39. Zack K

    Using Trig Substitution in Trig Integration

    Homework Statement Integrate: $$\int \frac{dx}{x^2\sqrt{4-x^2}}dx$$ Homework EquationsThe Attempt at a Solution I got to the final solution ##\int \frac{dx}{x^2\sqrt{4-x^2}}dx=-\frac{1}{4}cot(arcsin(\frac{1}{2}x))##. But It's the method where you transform that to the solution...
  40. ChristinaMaria

    Improper integral with substitution

    Hi! I am trying to solve problems from previous exams to prepare for my own. In this problem I am supposed to find the improper integral by substituting one of the "elements", but I don't understand how to get from one given step to the next. Homework Statement Solve the integral by...
  41. karush

    MHB 7.3.5 Integral with trig substitution

    $\textsf{Evaluate the integral}$ $$I=\displaystyle\int\frac{x^2}{\sqrt{9-x^2}}$$ $\textit{from the common Integrals Table we have}$ $$\displaystyle I=\int\frac{u^2}{\sqrt{u^2-a^2}} \, du =\frac{u}{2}\sqrt{u^2-a^2}+\frac{a^2}{2} \ln\left|u+\sqrt{u^2-a^2}\right|+C$$...
  42. Y

    MHB Substitution Method to solve linear simultaneous equation

    What I have done: I changed all fractions to common denom and that gave me 5y-5x=1 (1) *I numbered the fractions 5y+2x=5 (2) Then: 5y=5-2x Substitute into equation 1 (5-2x)-5x=1 5-7x=1 x=4/7 Thing is my answer says I should be getting x=0 Any hints?
  43. B

    A Question about computing residues after substitution

    Hi members, See attacged PDF file for my question Thank you
  44. donaldparida

    B Methods of integration: direct and indirect substitution

    I have seen two approaches to the method of integration by substitution (in two different books). On searching the internet i came to know that Approach I is known as the method of integration by direct substitution whereas Approach II is known as the method of integration by indirect...
  45. binbagsss

    Integrating with Substitution: Clarifying the Contour and Chain Rule

    Homework Statement Homework Equations below The Attempt at a Solution I have shown that the first identity holds true. Because this is true without it being surrounded by an integral I guess you would need to integrate it all around the same contour ##C##. So say I have: ## _C \int...
  46. Lazy Rat

    Integration problem using u substitution

    Homework Statement ## \int {sin} \frac{\pi x} {L} dx ##Homework Equations u substitution The Attempt at a Solution If i make ## u = \frac{\pi x} {L} ## and then derive u I get ## \frac {\pi}{L} ## yet the final solution has ## \frac {L}{\pi} ## The final solution is ## \frac {L}{\pi} - cos...
  47. T

    MHB Understanding Integral Substitution: Finding Equivalent Ranges for Functions

    Hi, I posted a question here a few days ago regarding some questions I've been doing on an online quiz. I seem to be getting stuck on the integral substitution questions. I've been slowly making progress, but some of these questions have been confusing me, and reading up on them is only giving...
  48. F

    Substitution in a differential equation, independent variable

    Homework Statement $$y'=-\frac{1}{10}y+(cos t)y^2$$ when doing substitute for ##z=\frac{1}{y}## I understand this is ##z(t)=\frac{1}{y(t)}## I know t is independent variable and y is dependent variable but I want to know what is z role here, is it change the dependent variable? when...
  49. T

    MHB How can substitution make solving integrals easier?

    Hi, I've got this problem that I've been trying to work out. I think most of my problems come from the fact that I am not yet well versed in u substitution when it comes to integrals. I'm also not 100% sure what the problem is asking. I've tried doing a couple of things, but they don't seem to...
  50. Wrichik Basu

    A query in integration using method of substitution

    Homework Statement :[/B] I was learning the use of standard forms in method of substitution in solving integration. My book has given this method for solving integrals of the type ##\int \frac{lx +m}{ax^2+bx+c} dx##: As an example, the book gives this one: Homework Equations :[/B] The...
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