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. R

    How to do this integral with substitution?

    This is the integral: NB everything in the expenential is in a squared bracket, couldn't get tex to do it \frac{1}{2\pi}\int_{\infty }^{\infty} e^{\tfrac{q\Delta}{\sqrt{2}}-\tfrac{ix}{\sqrt{2}\Delta}}dq The only information the tutor has given use to solve this is to use substitution and this...
  2. F

    MHB How to solve a differential equation using substitution?

    Solve the DE using an appropriate substitution. (x-y)dx+xdy=0 First step is to determine the substitution. I was told for homogeneous ODEs to always make the substitution y=ux but the substitution u=x-y looks better. Let u = x-y then u'=-y' which means y'=-u' rewrite the original equation...
  3. F

    MHB Solving Substitution Steps: Examples Explained

    In the context of "solutions by substitutions" in the examples there's a step I don't understand: What is going on in these two examples? If y=ux then \frac{dy}{dx}=u and dy=u dx \neq udx + xdu Same for the other, if y=u^{-1} then \frac{dy}{dx}=0?
  4. KleZMeR

    Complex Integral Trigonometric Substitution

    I'm solving two different definite integrals of functions \frac{sin(z)}{z} and \frac{cos(z)}{e^z+e^{-z}} with complex analysis and the residue theorem, and in the solutions they replace both sin(z) and cos(z) with e^{iz} why is this possible?
  5. L

    Differential Equations Help, non-linear first order with substitution

    (r^2) (dT/dr)+B*r*T=T^2, with initial condition dT/dr |r=0 =0 where B is a constant I've gotten it to this: dT/dr = -BT/r + T2 / r2 by dividing everything by r2, then I substitute using λ= T/r which gives: r * dλ/dr + lambda = -B * (λ) + λ^2 I don't know how to separate...
  6. R

    Trig Integration By Substitution

    Mod note: Moved from technical math section ∫(2x+6)/sqrt(5-4x-x^2) I have 2/3(ln|tan(theta)+sec(theta)|-3|cos(theta)|) where x=sin^-1((x+2)/3)
  7. N

    How to prove solution to Fick's second law by substitution

    Homework Statement I am to prove that a solution to the differential equation Fick's second law is valid by substitution. Homework Equations Fick's second law: \frac{\partial C}{\partial t} = \frac{\partial}{\partial x} \left( D \frac{C}{\partial x} \right) Solution to Fick's second...
  8. P

    Integration by substitution for sin/cos products

    Ok so I might be doing something silly but I just don't understand what is going on here. So the integral: i = ∫ sin x (cos x)^3 dx First I say u = cos x. So du = - sin x dx. So now I have i = ∫ - u^3 du. Which gives: i = -(1/4)u^4 or -(1/4)(cos x)^4. Easy. But if I say u = sin x...
  9. A

    Need help using substitution before integration

    Hi I just started calc 2 and am stuck on a problem. integral ln(x+x^2)dx. They want to use substitution prior to integrate by parts, but I'm completely stuck. Can anyone help explain how to solve?
  10. Dethrone

    MHB Why do the answers differ by a factor of 7?

    $$\int \frac{x}{\sqrt{x^2-49}}dx$$ Method 1: (by inspection) $$\int \frac{x}{\sqrt{x^2-49}}dx=\int \,d\sqrt{x^2-49}=\sqrt{x^2-49}+C$$ Method 2: (Trig substitution) $$\int \frac{x}{\sqrt{x^2-49}}dx=\int \frac{7\sec(\theta)}{\sqrt{49\sec^2(\theta) -49}}d(7\sec(\theta))=7\int...
  11. B

    How Does the Denominator x+a Become 1-t in Integral Substitution?

    Hi members, see attached Pdf file. For a<0,write b=-a and let x=bt.Then My question: how becomes the denominator x+a to 1-t? I don't see it Thank you
  12. A

    MHB Calculus I Simplifying and Substitution

    Hey guys, I have a couple more questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help. Question: For the first one, I evaluated by replacing x for 2+h for f(x). Then I substituted into the givern expression and simplified to get...
  13. Dethrone

    MHB Why Are My Tangent Half-Angle Substitution Answers Different from the Workbook?

    I did two questions from my workbook that involved the tangent half-angle substitution, z = tan (\frac{x}{2}). The answers that I got, for two questions, were different (but correct, I think) in the same way. Can you assist me in how to acquire the workbook answer? 1. \int \frac{dx}{1-2sinx}...
  14. A

    MHB Quick Trigonometric Substitution Question

    Hey guys, I'm really doubting my answer for 4b specifically. I used x=tan(Ø) and got (-1/4)cot(Ø) - (Ø) + C. I'm really not sure about this one. Thanks in advance.
  15. M

    Use Gaussian elimination with back substitution?

    Use Gaussian elimination with back substitution to solve the following system: x1+x2+x3=1, x1+2x2+2x3=1, x1+2x2+3x3=1. The answer is (1, 0, 0) and I know how to solve the problem but I just don't know if I should use bracket or the big parentheses for this type of problem when I solve...
  16. D

    DES Substitution Example and Homework Solution

    Homework Statement example: S(18 09 12 3d 11 17 38 39) = 5fd25e03 Homework Equations The Attempt at a Solution So I know about DES that you split a 64 bit block into left and right halves 32 bits each. Where even bits are on the left and odd are on the right. Perform the round...
  17. Dethrone

    MHB When to use a hyperbolic trig substitution in integration problems?

    I read somewhere that: sqrt(a^2-x^2), you can use x = asinx, acosx sqrt(a^2+x^2), you can use x = atanx (or acotx), asinhx sqrt(x^2-a^2), you can use x = asecx (or a cscx), acoshx When would it be beneficial to use a hyperbolic trig substitution as oppose to the regular trig substitutions (sin...
  18. H

    MHB Solving System of Equations using Substitution or Elimination

    x^2-13xy+12y^2=0 (1) x^2+xy=156 (2) What I have so far: x^2+xy=156 xy=156-x^2 y=(156-x^2)/x) Plugged y=(156-x^2)/x) into (1): x^2-13x(156-x^2)/x)+12(156-x^2)/x)^2=0 For 1st half I Multiplied x to -13x in order to get the same denominator so I can multiply it to (156-x^2)/x)...
  19. S

    MHB Solving an Integral Using Trig Substitution

    \int \frac{81/100}{100x^2 + 81} so I know this is 1 + tan^2\theta but how do I implement it? isn't it a^2 + x so should i let x = 100tan\theta?
  20. P

    Isopentane Cl2 substitution reaction

    Hello, I have a question. Let's say we have isopentane and Cl2 in a substitution reaction. How many different Carbon atoms can chlorine atom bond with? All five or can it substitute with a hydrogen that bonded to methyl brach? It is an easy question maybe but please answer that is really...
  21. S

    How do I do this integration by substitution?

    Homework Statement ∫1/(3+((2x)^.5))dx the answer should be ((2x)^.5) - 3ln(3+((2x)^.5)) + c I keep getting ((2x)^.5) - ln(3+((2x)^.5)) + c Homework Equations ∫1/(3+((2x)^.5))dx The Attempt at a Solution I did: u = 3 + ((2x)^.5) du = 1/((2x)^.5) dx du((2x)^.5) = dx...
  22. P

    Integration by substitution: Can I treat this as constant

    I am trying to compute the following integral: \int \exp^{w^T \Lambda w}\, d\theta where \Lambda is a constant wrt \theta w = y - t(x, \theta) So, I am trying to use substitution and I have: d\theta = \frac{-dw}{t^{'}(x, \theta)} So, substituting it, I have the following integral...
  23. P

    Trigonometric substitution of (x^2+8x)

    Homework Statement Hello PF, I am taking calculus II right now, and a homework problem I came to ponder upon has been giving me big trouble today. Here is the what I have to take the integral of: ∫x/(x^(2)+8x)^(1/2) dx Every other trig substitution problems were straight forward, as all...
  24. T

    Is this valid when doing u substitution for integration?

    So I'm doing length of an arc in my calculus 1 class. After plugging everything in the arc length formula. Now I have this complicated function to integrate. Square root of (16x^8+8x^4+1)/16x^4. I took the denominator out of my square root and got 4x^2. Now I take u=4x^2. Du/2x =dx...
  25. M

    Using u substitution for integrating.

    So I am pretty bad at u substitution. I don't really get how to replace values with du or u. Can you please give me tips on how to do u substitution well? Thanks.
  26. J

    Using trig substitution or partial fractions?

    When would you use trig substitution vs. partial fractions? I know partial fractions is when you have a polynomial over a polynomial, but some of the problems in the trig substitution section in my book had polynomial over polynomial and used trig substitution?
  27. E

    Trig substitution integration?

    Homework Statement Integrate dx/((x^2+1)^2) Homework Equations Tan^2=sec^2-1 The Attempt at a Solution So I let x=tanx then dx=sec^2x Then plugging everything in; Sec^2(x)/(tan^2+1)^2 So it's sec^2/(sec^2x)^2) which is sec^2x/sec^4x Canceling out the sec^2 gives...
  28. A

    Solve DE Homework: y'/y + lny = sqrt(1-e^x)

    Homework Statement Use appropriate substitution to solve the differential equation y'/y + lny = sqrt(1-e^x) Homework Equations The Attempt at a Solution I thought of trying to substitute y=ux but didn't get any helpful results, any help or hints would be great.
  29. P

    Integration using substitution

    1. $$\int \frac{1}{1+e^x}\,dx$$ Homework Equations Substitution The Attempt at a Solution $$u=1+e^x$$ $$du=e^xdx$$ $$\int \frac{1}{u}\frac{1}{e^x}\,du$$ $$\int \frac{1}{u}\frac{1}{u-1}\,du$$ $$\int \frac{1}{u(u-1)}\,du$$ $$= \ln {|u^2-u|} = \ln {|(1+e^x)-(1+e^x)|} = \ln...
  30. D

    Differential equation (substitution)

    Homework Statement for this question , i 've got my positive 9 but i got -64 , can anyone tell me which part is wrong? Question : https://www.flickr.com/photos/123101...3/13907725466/ Wroking : https://www.flickr.com/photos/123101...n/photostream/ Homework Equations The...
  31. A

    Maclaurin Series using Substitution

    Homework Statement Use a known Maclaurin series to compute the Maclaurin series for the function: f(x) = x/(1-4(x^2))Homework Equations 1/(1-x) = ∑x^nThe Attempt at a Solution I tried removing x from the numerator for: x ∑ 1/(1-4(x^2)), which would end up through substitution as x ∑...
  32. R

    Seemingly simple multiple integral substitution

    Homework Statement This problem seems fine but has me stumped for some reason... For the variables u=x2−y2 v= x y Under the condition u>0 We're simply asked to determine x(u,v) and y(u,v) so that we can eventually use them to solve a multiple integral by substitution. Homework Equations...
  33. MarkFL

    MHB Integrate sqrt(1-x^4)/x^5 dx Using Trig Sub | Yahoo Answers

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  34. D

    Successive Substitution vs Newton's Method

    Homework Statement Name disadvantages of the Successive Method vs Newtons for solving nonlinear equations? Homework Equations The Attempt at a Solution I went all through the textbook and this is all I could find on the successive method disadvantages but these are not compared to...
  35. C

    Using Integral Substitution to Solve a Challenging Integration Problem

    Homework Statement sorry if question is unclear can't draw the integal sign out Show that Integral infinity-0 dz/((e^2z) - 1)^1/2 = integral 1- 0 dx/(1-x^2)^1/2 = pi/2 The Attempt at a Solution I can get from the second integral to pi/2, as the second integral is sin^1(1) =...
  36. F

    Simple Substitution Taylor Polynomials

    I've been taught that with the basic form of a function's maclaurin series, complex forms of the same series can be found. For example, the first three terms for arctan(x) are x-x^3/3 + x^5/5, meaning the first three terms for arctan(x^2+1) at a=0 should be (x^2+1) - ((x^2+1)^3)/3 +...
  37. E

    Integration by substitution question

    Homework Statement I've been working on a problem from Apostol "Calculus" Volume 1 (not homework but self study). The problem is Section 5.8, Number 25 (Page 217) and states: If [tex]$\m$[\tex] is a positive integer, show that: \int_0^{\frac{\pi}{2}} cos^m x sin^m x dx =...
  38. M

    Integrating dx/(2+sin(x)) using a complex substitution

    Homework Statement Compute the real integral \int\frac{dθ}{2+sin(θ)}, where the limits of integration are from 0 to 2π by writing the sine function in terms of the exponential function and making the substitution z=e^{iθ} to turn the real integral into a complex integral. Homework...
  39. N

    Use a substitution to compute the integral

    Homework Statement Going over past exam problems, stuck on this one. Attached Calc 2, topics include for this exam integration techniques, such as partial fractions, improper integrals, trig sub, and series. Question reads: Use a substitution to compute: (see attached) Homework...
  40. S

    Use u = y' substitution to solve (y + 1) y'' = (y')^2

    Homework Statement Solve the following differential equation, by using the substitution u = y'.: (y + 1) y'' = (y')^2 Homework Equations I'm assuming: Chain Rule The Attempt at a Solution My problem is, simply, that I don't get how to go from u = y' to y'' = u du/dy, and I would...
  41. N

    Double Substitution Correct or Faulty?

    Hey. I was reviewing some trig substitution and I know how to solve the problem but I was curious if you can also solve it by using two different substitutions/change of varaibles? If so, how would you relate the functions? Consider for example: ∫ (1-x^2)^1/2 dx, -1<= x <= 1 x = sin Θ dx...
  42. G

    Different solutions to x/(1-x) by substitution and decomposition

    Homework Statement Derive x/(1-x) Homework Equations By substitution: u = (1-x) du = -dx ∫x/(1-x)dx = -∫(1-u)/u du = ∫1-(1/u)du = u-log(u)+c = (1-x) - log(1-x) + c By decomposition: x/(1-x) = 1/(1-x)-1 ∫1/(1-x)-1dx = -log(1-x)-x+c The Attempt at a Solution Which solutions...
  43. S

    How Accurate Is the Substitution in This Double Integral Problem?

    Hey. Homework Statement ∫∫x^3 dxdy, with the area of integration: D={(x,y)∈R^2: 1<=x^2+9y^2<=9, x>=3y} The Attempt at a Solution Did the variable substitution u=x and v=3y so the area of integration became 1<=u^2 + v^2 <=9, u>=v. And the integral became ∫∫(1/3)u^3 dudv. Then I switched to...
  44. 9

    2x + y = 8, x - z = 2. Solve by elimination and substitution

    Homework Statement 2x + y = 8, x - z = 2. Solve by elimination and substitutionHomework Equations 2x + y = 8, x - z = 2 The Attempt at a Solution It cannot be solved by elimination or substitution.
  45. jdawg

    Solving U-Substitution Problem with 2x√(2x-3) in Calculus

    Homework Statement ∫2x√(2x-3) dx Homework Equations The Attempt at a Solution u=2x du=2 dx 1/2∫u√(u-3) du Am I on the right track with this? I'm not really sure what to do next.
  46. S

    Help with Trig Substitution Integral Problem

    Homework Statement Question is attached in this post.Homework Equations Question is attached in this post. The Attempt at a Solution I've solved the problem via using x=asinθ where a=1 I've been able to integrate the problem to the point where I get cos^2(θ)/sin^2(θ), but can't seem to...
  47. S

    Help with Volume of Revolution/Trig Substitution Problem

    Homework Statement The problem is attached in this post. Homework Equations The problem is attached in this post. The Attempt at a Solution Disk method with the radius equal to x/((x^2+3)^5/4) For Trig Substitution √(x^2+a^2) -> x=atanθ a=√3 -> a^2=3 x=√(3)tanθ -> dx=√(3)sec^2(θ)...
  48. S

    Substitution and Integration by Parts

    Homework Statement First make a substitution and then use integration by parts to evaluate the integral. ∫x^{7}cos(x^{4})dx Homework Equations Equation for Substitution: ∫f(g(x))g'(x)dx = ∫f(u)du Equation for Integration by Parts: ∫udv = uv - ∫vdu The Attempt at a Solution So...
  49. S

    MHB What is the purpose of substitution in integration?

    Suppose we have \int \frac{4}{x^2 + 4} So I understand the first thing we would so is bring the constant out and do u substitution but what I don't understand is how we can make the substitution u = \frac{x}{2} when there clearly is no \frac{x}{2} in the problem. I also understand how to...
  50. T

    Trig substitution into integrals

    I was testing for convergence of a series: ∑\frac{1}{n^2 -1} from n=3 to infinity I used the integral test, substituting n as 2sin(u) so here's the question: when using the trig substitution, I realized the upperbound, infinity, would fit inside the sine. Is it still possible to make...
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