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. T

    Finding Points of Intersection by Substitution

    Homework Statement Find any points of intersection of the graphs by the method of substitution. xy+x-2y+3=0 x^2+4y^2-9=0 Homework Equations The Attempt at a Solution From the second equation I can solve for y: y=\frac{\sqrt{9-x^2}}{2} Plug it into the first equation and...
  2. C

    Integrals involving trig substitution

    Homework Statement I don't know how to solve these. How do you evaluate the integral of \frac{3dx}{\sqrt{3+X^2}}? I know you have to set x=atan\theta. our a is \sqrt{3} so x =\sqrt{3}tan\theta. That means dx=\sqrt{3}sec2\thetad\theta. I also made a right triangle using the information...
  3. D

    Integration by trig substitution

    Homework Statement the problem is INTEGRAL 6dz/(z^2(sqrt(z^2+9)) z^2+ a^2 , then z=a*tan@ where a here is 3, because 3^2=9 , i use @ here to represent theta substituting this for z; Int[6/(z^2(sqrt(z^2+9))]dz= Int[(6*3sec^2@d@)/(9tan^2@(sqrt(9tan^2@+9))]=...
  4. D

    Integration by substitution where square root is U^2

    Hi there, I am having difficulty with one aspect of intergration by substitution where the substituion of a square root is U^2, wondering if anyone can help. Problem: Integral of: 2x√(3x-4) dx by substituting U^2 = 3x-4 Would du^2/dx = 3 therefore 1/3 du^2 = dx (I think...
  5. rubenvb

    Double integral to single by magic substitution

    double integral to single by "magic" substitution Hi, I have a double (actually quadruple, but the other dimensions don't matter here) integral which looks like this: \iint_0^\infty \frac{d^2 k}{k^2} Now, someone here told me to replace that with \int_0^\infty \frac{1}{2} 2\pi...
  6. G

    Solve second order diff equation using substitution

    Homework Statement d2y/dx2-dy/dx+y*exp(2x) = x*exp(2x)-1 substitute t=exp(x) and set z(t)=y(x) and rewrite hence find all solutions The Attempt at a Solution Rewriting gives: d2z/dt2-dz/dt+z*t^2=(ln(t) * t^2) - 1 however I don't see how this in any way helps us...
  7. T

    Solve ODE by Substitution: Find General Solution

    Homework Statement By means of substitution x=X+1, y=Y+2 ,shwo that the equation dy/dx=(2x-y)/(x+2y+5) can be reduced to dY/dX=(2X-Y)/(X+2Y).Hence, find the general solution of the given equation. Homework Equations The Attempt at a Solution The first part is quite simple to...
  8. Telemachus

    Solving an integral using a*sinh substitution

    Homework Statement The statement says: Calculate the next integrals using the adequate trigonometric substitution: \displaystyle\int_{}^{}x^2\sqrt[ ]{x^2+3}dxHomework Equations ch^2(t)-sh^2(t)=1\Rightarrow{ch(t)=\sqrt[ ]{1+sh^2(t)}}The Attempt at a Solution x=\sqrt[ ]{3}sh(t) dx=\sqrt[...
  9. Telemachus

    Solving an integral using an special substitution

    Homework Statement Well, the exercise asks me to solve the next integral using an adequate substitution. \displaystyle\int_{}^{}\sqrt[ ]{4-x^2}dx The Attempt at a Solution \displaystyle\int_{}^{}\sqrt[ ]{4-x^2}dx What I did was: x=2\sin\theta dx=2\cos\theta d\theta So, then I get...
  10. Telemachus

    Solving an integral by substitution method

    Homework Statement Hi there. I'm dealing with undefined integrals now. And I found this one that I don't know how to solve. The problem statement says: Solve the next integrals using the substitution method. \displaystyle\int_{}^{}\displaystyle\frac{\cos(x)}{\sin^3(x)}The Attempt at a Solution...
  11. 2

    Hope this helps!Best regards,Nalin Pithwa

    Hi, I've been doing some additional maths papers and I've seen the use of the substitution u=tan(x/2) in order to calculate integrals. In the mark scheme it states that this particular substitution used to be fairly common, however is not on the modern A-level syllabus. Would someone...
  12. T

    What is the Correct Integration by Substitution for \int \frac{3x}{2x+3}?

    Homework Statement \int \frac{3x}{2x+3} u = 2x +3 x = \frac{1}{2}(u-3} ) dx = \frac{1}{2} du so now the integral should be, \int \frac{ \frac{3u-9}{2}}{u} \times \frac{1}{2} du = \frac{1}{2} \int \frac{3u-9}{2} \times \frac{1}{u} du \frac{1}{2} \int...
  13. L

    Solving Integrals Using Substitution: A Step-by-Step Guide

    I'm attempting to solve the following problem: \int_{0}^{\infty} {\frac{x arctan(x)}{(1+x^{2})^{2}}dx} I started with a substitution: u=arctan(x), du=\frac{1}{(1+x^{2})}dx This seemed like the right thing to do, but after trying to put it together in several different ways I got...
  14. L

    Trigonometric Substitution Problem

    This problem looks relatively simple, but the coefficient in front of the variable is causing issues: \int{\sqrt{1-4x^{2}}}dx So I started like this: x=sin(\theta) dx=cos(\theta)d\theta \int{\sqrt{1-4sin^{2}(\theta)}cos(\theta)d\theta} Normally you can remove the constant from the root and...
  15. stripes

    Problem with Integration by substitution

    Homework Statement If f is continuous and \int^{9}_{0}f(x)dx = 4, find \int^{3}_{0}xf(x^{2})dx Homework Equations None required The Attempt at a Solution Don't really know where to begin, but I tried: for \int^{3}_{0}xf(x^{2})dx let: u = x^{2} du = 2xdx substitute...
  16. W

    Solving Integral of e^6x with U Substitution

    Hey All, First post, hopefully it will be readable. I was going to try and word it correctly, but I might as well just post a problem I am having with a certain notation. Take integral of e^6x. Easy enough question. Using U substitution: u = 6x du/dx = 6 du = 6 dx Integral above...
  17. T

    Substitution homework problem

    Homework Statement Evaluate ∫ -2 to 2 (x + 3)(4 - x^2)^1/2 dx by writing it as a sum of 2 integrals and interpreting one of those integrals in terms of an area. Homework Equations None. The Attempt at a Solution ∫ -2 to 2 (x + 3)(4 - x^2)^1/2 dx = ∫ 0 to -2 (x + 3)(4 - x^2)^1/2...
  18. B

    Second order differential equation via substitution

    Homework Statement Substitute p = \frac{dx}{dt} to solve x\prime\prime + \omega^2x = 0 Homework Equations \frac{dp}{dx} = v + x\frac{dv}{dx} v = \frac{p}{x} The Attempt at a Solution p = \frac{dx}{dt}, \frac{dp}{dt} = \frac{d^2x}{dt^2} \frac{dp}{dt} = \frac{dp}{dx}\frac{dx}{dt} =...
  19. A

    What is the Appropriate Substitution for Solving the Integral of x/(x^2+2x+2)dx?

    integral of x/(x2+2x+2)dx first thing i did was complete the square to get x/((x+1)2+1 i tried then having x+1 = tanx but that didnt work out because of the x on top i can't just set w = x+1 what would the right substitution be? any hints or help would be appreciated
  20. M

    U substitution or substitution by parts?

    Homework Statement ∫〖e^√x/√x dx〗 would this be a u substitution or a substitution by parts? Homework Equations The Attempt at a Solution
  21. L

    Using u substitution, which of the following is equivalent to this integral?

    Homework Statement Using the u substutituion u = 2x + 1, ∫(2x + 1)1/2dx (when x goes from 0 to 2) is equivalent to? Answer: (1/2)*∫(u)1/2du (when x goes from 1 to 5) Homework Equations The Attempt at a Solution If u is 2x + 1, then du = 2dx. Thus, I get (1/2)*∫(u)1/2du...
  22. A

    Trigonometric substitution for integral with exponential and square root

    Homework Statement Evaluate \int\frac{e^t}{\sqrt{e^2^t+9}} Homework Equations N/A The Attempt at a Solution i'm using substitution tan \theta = \frac{e^t}{3} or i also can use tan \theta = \frac{3}{e^t} both will get the same answer. am i right? because my...
  23. M

    Indefinite Integrals & Substitution Rule

    Homework Statement 2. The attempt at a solution I
  24. E

    Evaluate the integral using substitution

    1. Evaluate the integral [0,ln(3)] of ff(x)=(e^2x + 1)^2 /e^x I am having trouble locating what to substitute.
  25. W

    Understanding Substitution in Differential Equations | Homework Help

    Homework Statement I'm reading a book where they do the following steps which I don't understand: We have a DE: b^2 * y'' = axy put t = b^(-2/3) a ^(1/3) x then somehow get (d^2 y)/dt^2 = ty how? Homework Equations None. The Attempt at a Solution I tried messing with chain...
  26. T

    Partial fractions & Substitution Integration

    Homework Statement Hi, \int \frac{1}{x(x^{2}+1)}dx Homework Equations The Attempt at a Solution well I split this into partial fractions \frac{A}{x} + \frac{Bx + C}{x^{2} + 1} so 1 \equiv A(x^{2}+1) + (Bx + C)x when x = 0, A =1 when x = 1, Bx + C = -1 so...
  27. T

    Find the Integral Using Substitution: \int \frac{-2 \sqrt{1-x}}{2 + \sqrt{1-x}}

    Homework Statement By making the substituion t = \sqrt{1-x} find \int \frac{1}{2 + \sqrt{1 - x}}Homework Equations The Attempt at a Solution So t = (1-x)^\frac{1/2} t' = - \frac{1}{2} (1 - x)^{-\frac{1}{2}} dx = -2 \sqrt{1-x} dt \int \frac{-2 \sqrt{1-x}}{2 + \sqrt{1-x}} dt \int...
  28. 0

    U-Substitution for Indefinite Integrals

    Hi, am I on the right track with this U-substitution problem? Homework Statement Evaluate the indefinite integral Homework Equations integral of x^2(x^3 + 5)^9 dx The Attempt at a Solution integral of x^2(x^3 + 5)^9 dx Let u = x^3 + 5 du = 2x^2 1/2du = x^2 1/2 integral u^9 du 1/2...
  29. K

    Double Integral Substitution Techniques for Evaluating Complex Integrals

    Homework Statement Evaluate the integral. 1|0 s|0 ( t . sqrt ( t2 + s2 ) dt dsI hope the way I've written it makes some sort of sense. The Attempt at a Solution After getting my head around changing the order of integration I get hit with this question and for some reason am totally...
  30. S

    Simple Trigonometric Substitution Problem

    Homework Statement \int{\frac{x^{3}}{\sqrt{4 - x^{2}}}} NOTE: The use of "0" is theta. I couldn't figure out how to insert one :\ Homework Equations The trig identity sin^{2}0 = 1 - cos^{2}0. The Attempt at a Solution I thought I completed the problem fine, but I realized WolframAlpha has a...
  31. N

    Trigonometric substitution

    \int\frac{x}{\sqrt{x^2+x+1}}dx \int \frac{x}{\sqrt{(x+\frac{1}{2})^2+\frac{3}{4}}}dx u=x+\frac{1}{2} \int \frac{u-\frac{1}{2}}{\sqrt{u^2+\frac{3}{4}}}du u=\frac{\sqrt{3}}{2}tanT du=\frac{\sqrt{3}}{2}sec^2TdT \int...
  32. T

    Eigenvalue Factorization and Matrix Substitution

    In my literature reviews I found a few things that I can't quite understand. Homework Statement I have the following equation: http://img717.yfrog.com/img717/6416/31771570.jpg I'm told that by using the eigenvalue factorization: http://img89.yfrog.com/img89/760/83769756.jpg , I can...
  33. N

    Trigonometric substitution

    9x^2-4y^2=36 \frac{x^2}{4}-\frac{y^2}{9}=1 y=\frac{3}{2}\sqrt{x^2-4} 3\int_{2}^{3}\sqrt{x^2-4}dx x=2sect dx=2secttant 12\int_{a}^{b}tan^2tsectdt 12\int_{a}^{b}(sec^2t-1)(sect)dt 12\int sec^3tdt-12\int sectdt 6\int secttant-6\int ln|sect+tant|...
  34. DocZaius

    Method for finding non-obvious substitution in integration

    To find the integral of the sec(x), you have to substitute a term that is not immediately obvious. \int sec(x) dx = \int sec(x) \frac{sec(x)+tan(x)}{sec(x)+tan(x)} dx u= sec(x)+tan(x) du= (sec(x)tan(x)+sec^{2}(x))dx \int sec(x) \frac{sec(x)+tan(x)}{sec(x)+tan(x)} dx = \int...
  35. N

    Elimination vs substitution & ethanol as a solvent

    Hello! I was looking into haloalkane reactions and the factors which determine the proportion of nucleophilic substitution to elimination reactions. I read that ethanol is more conducive to elimination reactions than substitution reactions, it mentions it being less polar than water, which...
  36. I

    Webpage title: Solving Integrals Using Substitution Method

    Homework Statement Question is:Integrate x(2x+1)^8 dx in terms of x. Homework Equations The Attempt at a Solution Here is how i started off:by relabeling them. let u = 2x+1. du/dx = 2. dx=du/2. Also x=u-1/2. So my terms now are: Integral (u-1/2)u^8 (du/2) <- this is where i...
  37. N

    Find the integral of x/(x-6) with substitution

    \int\frac{x}{x-6}dx u=x-6 \int \frac{u+6}{u}du \int 1+\frac{6}{u} du u+6ln|u|+C x-6+6ln|x-6|+C this appears to be incorrect although it seems logical, also if somone could please tell me the syntax for the definte integral also similar case here \int \frac{x^2}{x+4}dx...
  38. 3

    Solving Substitution Integrals: Guide and Example Problems

    Homework Statement \int(x^5\sqrt{x^2+4})dx The answer is given as: =105(x^2+4)^\frac{3}{2}(15x^4-48x^2+128)+C Homework Equations The Attempt at a Solution u=\sqrt{x^2+4} u^2=x^2+4 2udu=2xdx udu=xdx u^2-4=x^2 \int(x^5\sqrt{x^2+4})dx = \int(u^2+4)^2u^2du =...
  39. James889

    Inverse trig substitution integral

    Hi, I need to integrate the following: \int \frac{x^2}{\sqrt{9-x^2}} So let x = 3sin\theta \frac{dx}{d\theta} = 3cos\theta So i now have the integral of \frac{9sin^2\theta \cdot 3cos\theta}{3cos\theta} How do i go about the integration from here? parts?
  40. S

    Trigonometric Substitution- Area help

    Homework Statement Let R be the smaller of the two regions enclosed by the elipse 144 x2+64 y2=9216 and the line $ x=(8 \sqrt{2})/2$. Find the area of the region R. Homework Equations The Attempt at a Solution my textbook doesn't have anything like this.. i have no idea where...
  41. B

    Solving Integrals with Trig Substitution - 1/(25-x^2)

    Hi Everyone! I just need some guidance on this problem. I seem to have trouble what integration technique I need to use on integrals of this type. Homework Statement integrate 1/(25-x^2) Homework Equations sqrt(a^2-u^2) arcsin(u/a) The Attempt at a Solution Would I be...
  42. C

    Evaluate integral using substitution

    Homework Statement evaluate using substitution Integral [cos^-1 x]/sqrt[1-x^2] dx Homework Equations The Attempt at a Solution I am just starting with integration and I am getting frustrated with this problem. If someone could show me how to setup and start this problem so I could...
  43. B

    Trig Substitution: Solving Integrals with sec^3Θ

    http://img708.imageshack.us/img708/8897/symimage.gif so I did x=atanΘ. which is x=3tanΘ and dx is 3sec^2\Theta. Then it is \sqrt[]{9tan^2\Theta+9}*3sec^2\Theta which evaluates after factoring to \sqrt[]{9sec^2\Theta}*3sec^2\Theta which is then 3sec\Theta*3sec^2\Theta If i take the 9...
  44. W

    Integration With Trig Substitution Calc II

    Homework Statement \int \frac{\sqrt{196 x^2-144}}{x} dx Homework Equations The Attempt at a Solution I first rewrote the integral... \int \frac{\sqrt{(14x)^2-12^2}}{x} dx Then I let... 14x=12sec\theta thus... x=6/7sec\theta dx=6/7sec \theta tan \theta d \theta My...
  45. M

    My new U substitution approach? is this legal?

    Allow me to explain my new theory, The "Mancini conjecture." Ok...lets say I have an integral like (4-x^2)^(1/2) dx. and letting u = 4-x^2, we get du/dx = -2x, and if I took the second derivative of du/dx...i would get -2 this would be ideal, because I would then have du'' = -2 dx, or -1/2...
  46. R

    Trig Substitution (?) Integral

    Homework Statement The answer is: The Attempt at a Solution I tried trig substitution, letting x =\sqrt{2}tan(\theta) and using the identity 1+tan^{2}=sec^{2}(\theta), but couldn't get to the answer. Thanks for the help.
  47. M

    How can I evaluate this integral using trig substitution?

    Homework Statement evaluate the integral using the indicated trig substitution. sketch the corresponding right triangle. integral of(1/(x^2 sqrt(x^2 - 9)) Homework Equations integral of(1/(x^2 sqrt(x^2 - 9)) The Attempt at a Solution at first glance this seemed really easy...
  48. 3

    Integration using Trig. Substitution

    Homework Statement \int\sqrt{X^2+1}dX Homework Equations The Attempt at a Solution I used the substitution X=tan \theta So, dX=(sec^2 \theta) d\theta Substituting in for X, I get: \int\sqrt{(tan^2 \theta)+1}(sec^2 \theta) d\theta = \int\sqrt{(sec^2 \theta)}(sec^2...
  49. R

    Integration by Parts substitution

    Homework Statement \int\arctan(4t)dt Homework Equations The Attempt at a Solution \int\arctan(4t)dt = t\arctan(4t) -4 \int \frac{t}{1+16t^2}dt I'm stuck at this point. I think I need to make a substitution for the denominator, but I'm not sure how to go about doing so.
  50. B

    Forward substitution in this case? Is it as simple as I think it is?

    Hi everyone: I am very rusty on linear algebra, so apologies if this is a silly question. The question is, in the system below, is it correct to take the calculated value of uik+1 from each PREVIOUS step and simply plug it in at the NEXT step where (a * ui-1k+1 is required. I need to...
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