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. C

    Proving Definite Integral Using Substitution | Solving Math Problem

    Homework Statement prove by substitution that definite integral int (1/t)dt from [x to x*y] = int (1/t)dt from [1 to y]. Homework Equations The Attempt at a Solution i can do this problem if i integrate and use the log laws, no probs, but the question says to use a substitution...
  2. C

    Development of Integration by Substitution

    How was Integration by Substitution and Trig Substitution developed? My calc book doesn't have much info, just a short (not really complete) proof. Could someone explain and/or lead me in the right direction?
  3. R

    Important intergration substitution

    int^{0}_{t}[cos(sqrt{x}]dx can anyone tell me the solution to this question !
  4. A

    Integration by Trigonometric Substitution.

    I'm not sure about answer.It looks very strange. Homework Statement \int_{1}^{e}\frac{dx}{x\sqrt{1+ln^2x}} The Attempt at a Solution for u=lnx-->u'=1/x \int \frac{du}{\sqrt{1+u^2}} substituting u=tan\theta =\int \frac{d\theta}{cos\theta}=ln|sec\theta+tan\theta|...
  5. tony873004

    Solve Trig Substitution Integral: \int_{}^{} {\frac{x}{{\sqrt {3 - x^4 } }}dx}

    [SOLVED] trig substitution This is from the class notes. Evaluate the integral: \int_{}^{} {\frac{x}{{\sqrt {3 - x^4 } }}dx} \begin{array}{l} u = x^2 ,\,\,du = 2x\,dx\,\, \Leftrightarrow \,\,dx = \frac{{du}}{{2x}} \\ \\ \int_{}^{} {\frac{{x^1 }}{{\sqrt {3 - x^4 } }}dx} =...
  6. M

    How do I integrate u^-1 with new limits after substitution?

    Homework Statement Homework Equations None. Well, dx=du/cosx The Attempt at a Solution I've substituted it in, got new values for the limits but I have u^-1 on the bottom and so can't integrate it from my current knowledge. Basically I'm stuck with: Integration of u^(-1) du...
  7. tony873004

    Where does the x/3 come from in trigonometric substitution?

    This is an example from the book. Evaluate \int {\frac{{\sqrt {9 - x^2 } }}{{x^2 }}dx} I understand all the steps that get me up to = - \cos \theta \, - \theta \, + C Then the book goes on to explain: "Since this is an indefinate integral, we must return to the original variable...
  8. C

    Substitution and integration by parts

    Homework Statement Can anybody help me integrate x^3 e^{x^2} The Attempt at a Solution I can't see how to do it by substitution or integration by parts.
  9. Y

    Trig Substitution for Solving Integrals: Step-by-Step Guide

    \int \frac{x^2}{\sqrt{9-x^2}} find the integral using trig sub x= 3 \sin {\phi} replace 3sin\phi into x and solve. I got to \int \frac{9-9 \cos{\phi}}{3 \cos{\phi}} then what should I do?
  10. L

    Integrating \sqrt{16-(2x)^{4}}xdx using Trigonometric Substitution

    \int\sqrt{16-(2x)^{4}}xdx Hint says you may like to use the identity sin(theta)cos(theta)= sin(2theta)/2 However, I think I found a way to use 1-sin^2(theta)=cos^2(theta) First, (2x)^4 = 16x^4 So make it 16(1-x^2)^2. Take the 16 out of the root and the integral and you have...
  11. P

    Tricky Integral ( with substitution)

    Homework Statement {\int_{}^{}}{ \frac{ds}{{({s}^{2}+{d}^{2})}^{\frac{3}{2}}}} s \equiv variable d \equiv constant Homework Equations u-substitution techniques for integration. The Attempt at a Solution This integral is particularly tricky as I have already made several...
  12. L

    Integration By Parts and Substitution

    [SOLVED] Integration By Parts and Substitution Short background; Took Calc 1 my senior year in high school. Got As all 4 quarters and found it quite easy. Freshman year comes around and I sign up for Calc 2. Turns out the only teacher teaching Calculus 2 for my fall and spring semester is a...
  13. R

    Is this a good substitution that will work

    Homework Statement Prove \int_0^{1} \frac{1}{\sqrt{x^2+6x+25}} = ln(\frac{1+\sqrt{2}}{2})Homework Equations The Attempt at a Solution \int_0^{1} \frac{1}{\sqrt{x^2+6x+25}} = \int_0^{1} \frac{1}{\sqrt{(x+3)^2+16}} Let x+3=4tan\theta so that dx=4sec^2\theta d\theta and so the problem becomes...
  14. Saladsamurai

    U Substitution: Solve \int sec^3(2x)tan(2x) - Casey

    So I have another U substitution. \int sec^3(2x)tan(2x) this one is a little tricky for me. I have tried letting u= sec2x and tanx and 2x. 2x definitley gets me nowhere. I may be mistaken on the others. I will recheck them. I was also thinking of rewriting it as \int sec^4(2x)sin(2x)...
  15. Saladsamurai

    What is the Derivative of Arctan?

    I know this must be similar... \int \frac{e^x}{1+e^{2x}} should u=1+e^{2x}? Casey
  16. G

    Integration, u substitution, 1/u

    [SOLVED] Integration, u substitution, 1/u -- +C at the end of the integral solutions, I can't seem to add it in the LaTeX thing -- Homework Statement #1 \int\frac{1}{8-4x}dx #2 \int\frac{1}{2x}dx The Attempt at a Solution #1 Rewrite algebraically: \int\frac{1}{x-2}*\frac{-1}{4}dx Pull out...
  17. C

    How Can Substitution and Integration Help Solve This Equation?

    Homework Statement /int (2t+4)^-1/2 dt. the answer is 2(sqrt3)-2 Homework Equations The Attempt at a Solution u^-1/2*(1/2)du (-1/2)(u^1/2) (-1/4)(2t+4)^1/2 (-1/4)(sqrt 12)= (-1/4)(4(sqrt 3)/1)=sqrt 3 (sqrt 3)-1/2 2(sqrt 3)-1
  18. V

    How Does SO3H Attach to Benzene in Electrophilic Substitution?

    The benzene are sulphonated using acid sufuric. Please show me how the substitution happened as i really don't see how SO3H can attached to the benzene group and how the SO3H are separated from H2SO4. I really need to understand this substitution.. Thanks
  19. B

    Integrating with Trigonometric Substitution: Solving ∫ x √ 4 + x2 dx

    Homework Statement Evaluate ∫ x √ 4 + x2 dx by using the trigonometric substitution x = 2tanθ I am starting on the right track by subbing x=2tanθ into x like this: =∫ 2tanθ √ 4 + 2tanθ(2) then, do I just integrate that for the correct answer?
  20. D

    How can I evaluate the integral using the substitution u=1/x?

    Using the substitution u=1/x, evaluate: \int {\frac{{dx}}{{x^2 \sqrt {1 - x^2 } }}} I was able to do it making the substitution x=cos\theta, but I am supposed to show a worked solution using the given substitution. \int {\frac{{dx}}{{x^2 \sqrt {1 - x^2 } }}} = \int {\frac{{ - x^2...
  21. Z

    Integrating with Trig Substitution: Solving \int_{-2}^2 \frac{dx}{4+x^2}

    \int_{-2} ^2 \frac{dx}{4+x^2} I use the trig substitution and get everything done but for some reason I can't get the answer, here's all my working: x = 2 \tan\theta dx = 2 \sec^2\theta 4+x^2=4(1+\tan\theta)=4\sec^2\theta \int \frac{2\sec^2\theta d\theta}{4\sec^2\theta} \int...
  22. C

    Algebraic Substitution: Solving F(r) Using r=r_+(1+\rho^2)

    Homework Statement I have this function F(r)=\frac{(r-r_+)(r-r_-)}{r^2} and I want to make the subsitution r=r_+(1+\rho^2). Homework Equations None. The Attempt at a Solution So, I sub in, to obtain...
  23. B

    Why Am I Getting Incorrect Solutions to This Logarithmic Substitution Problem?

    Homework Statement (Idk how to put in the equation to make sense, therefore it is at the link below) Homework Equations The Attempt at a Solution Here is all I have done. Something just isn't right...there should be 3 answers (in the back of the book) because there is a cube...
  24. A

    Explaining the Substitution Rule: Easier Terms for Allison

    Can someone please explain the Substitution Rule to me in easier terms? I am soooo confused! Thanks, Allison
  25. F

    Calculating 4 Integral: A Tan & Cos Theta Substitution

    Homework Statement how would one calculate 4 \int_0^{\frac{\pi}{2}} \frac{\cos^2 \theta}{(1 + \cos^2 \theta)^2} d \theta ? The Attempt at a Solution someone suggested a u = \tan \theta substitution, but i don't understand why and how this would help me. couldn't i just use u = \cos t?
  26. C

    Evaluating the Integral Using Trig Substitution

    The question is to evaluate the integral in the attachment. Using trig substition, I've reduced it to ∫ (tanz)^2 where z will be found using the triangle. I just need to integrate tangent squared which I can't seem to figure how to do. I tried using the trig identity (secx)^2 - 1 but I don't...
  27. N

    Integration (different substitution = different answer)

    Out of curiosity there are several trig functions that can be integrated (WITHOUT the use of trig identities) using Integration by Substitution. One particular example is this: sin(x)cos(x) dx Integrating this with substitution u = cos(x) works out fine. HOWEVER integrating with...
  28. C

    Substitution rule for vectorial functions

    You remember the substitution rule (or Change of variables theorem), when the integrand is some real function of real variable. I would like to know if that rule has a version when the integrand is some vectorial function (of real variable). Thanks for your attention.
  29. T

    Trig Substitution: Solve Int \sqrt{x^2 + 16}

    Homework Statement \int\frac{dx}{\sqrt{x^2 + 16}}Homework Equations x = 4\tan\theta dx = 4\sec^2\theta \ d\thetaThe Attempt at a Solution \int\frac{4\sec^2\theta}{\sqrt{16\tan^2\theta + 16}}\ d\theta \int\frac{4\sec^2\theta}{\sqrt{16(\tan^2\theta + 1)}}\ d\theta...
  30. U

    Complex substitution into the equation of motion.

    Homework Statement The equation of motion of a mass m relative to a rotating coordinate system is m\frac{d^{2}r}{dt^2} = \vec{F} - m\vec{\omega} \times (\vec{\omega} \times \vec{r}) - 2m(\vec{\omega} \times \frac{d\vec{r}}{dt}) - m(\frac{d\vec{\omega}}{dt} \times \vec{r}) Consider the case F =...
  31. S

    What is the integral of (x+1)/Square root(4-x^2)?

    Homework Statement Evaluate the following integrals or state that they diverge. Use proper notation. Integral from 0 to 2 of (x+1)/Square root(4-x^2) Homework Equations The Attempt at a Solution I just substituted x = 2sin(theta) thus dx = 2cos(theta) I got to the...
  32. J

    Simple Integration by substitution

    Homework Statement Find by letting U^2=(4 + x^2) the following \int_0^2\frac{x}{\sqrt{4 + x^2}}dx? I can solve it by letting \mbox{x=2} tan(\theta), But I want to be able to do it by substitution. The Attempt at a Solution...
  33. J

    Trig substitution (integration)

    Homework Statement Homework Equations The Attempt at a Solution I'm not asking for someone to do the question for me but I was just wondering what I'm supposed to sub in. Do I put in as if it was (x^2-9)^(1/2) or do I have to do something differently if there is a constant in front...
  34. O

    INtegration by substitution followed by parts:

    ... of (x tan^2 x). i don't know how to do it. pls help
  35. tony873004

    Solving the Int. of Sec^3x Tan x: What is Correct?

    Homework Statement \int {\sec ^3 x\,\,\tan x\,\,dx} Homework Equations u = \sec x This is my guess at u. The Attempt at a Solution \frac{{du}}{{dx}} = \sec x\,\,\tan x,\,\,\,dx = \frac{{du}}{{\sec x\,\,\tan x}} \int {\sec ^3 x\,\,\tan x\,\,dx} = \int {u^3...
  36. tony873004

    I really don't get the substitution rule

    I really don't get the substitution rule. This is supposed to be the easiest problem in the homework set: u=3x \int {\cos \,3x\,\,dx\,\, = \,\,\int {\cos \,u\,\, = \,\,\sin \,u + C\,\, = \,\,\sin 3x + C} } But the right answer is 1/3 sin(3x). Where did the 1/3 come from?
  37. U

    How do I use substitution to solve a differential equation with a square root?

    Homework Statement Solve the differential equation. dy/dx = 4x + 4x/square root of (16-x^2) Homework Equations Substituting using U... The Attempt at a Solution I'm not sure if that's what I am supposed to do, but I tried using the U substitution... 4x + 4x/square root of...
  38. J

    Simple differential equation substitution

    Hi! I am looking through some solved exercises. One of them is the following: Solve the equation: x^2 y'' + (x^2 - 3x)y' + (3-x)y = x^4 knowing that y=x is a solution of the homogeneous equation. The professor then solves it by doing the following substitution: y=xz. Then he...
  39. B

    Integration by substitution and by parts

    I did a few problems in integration by parts. There are two that I just can't seem to get. I've tried every type of subsitution or part I can think of. 1. e^sqrt(x) 2. sin (ln x)
  40. S

    Which Factors Determine the Rate of SN1 and SN2 Reactions?

    1) Predict the relative reaction times from fastest to slowest for the following compounds with NaI in acetone: 1-chlorobutane, 1-bromobutane, 2-chlorobutane. I am assuming that this is under SN2 reaction conditions with the solvent and compound given. 2-chlorobutane = secondary...
  41. A

    General Formula using Miscellaneous Substitution

    I'm really stomped with this problem... i can't seem to get the answer... anyway... here's the problem.. (2(x^3) - (y^3))y'=3(x^2)y and i need to get the general solution... SO, here's what i did... I Let u=x^3 and du=3x^2dx so what happens is 2udy-(y^3)dy= ydu and...
  42. mbrmbrg

    Integral with trig substitution

    The bit of the problem that I'm working on: 6\int\frac{dx}{x^2-x+1} My work: =6\int\frac{dx}{(x^2-x+\frac{1}{4})+1-\frac{3}{4}} =6\int\frac{dx}{(x-\frac{1}{2})^2+\sqrt{\frac{3}{4}}^2} let x-\frac{1}{2}=\sqrt{\frac{3}{4}}\tan\theta so dx=\sqrt{\frac{3}{4}}\sec^2\theta d\theta...
  43. S

    Integrating x√(x^2+a^2) using Substitution Method

    Hello, evaluate the following integral: \int x \sqrt{x^2+a^2}dx definite integral from 0 to a what I did was u = x^2 + a^2 du = 2xdx 1/2 sqrt(u)du I just dropped the a^2 because we were finding the derivative of x but feel that it's very wrong.Any suggestions are much appreciated. thanks.
  44. S

    Integration by substitution ((sin(x))/(1+cos^2(x)))dx

    evaluat the indefinite integral ((sin(x))/(1+cos^2(x)))dx I let u = 1 + cos^2(x) then du = -sin^2(x)dx I rewrite the integral to - integral sqrt(du)/u can I set it up like this? should I change u to something else? I also tried it like this by rewriting the original equation...
  45. S

    Indefinite integral substitution

    evaluate the indefinite integral ((e^x)/((e^x)+1))dx I let u = ((e^x)+1) then du = (e^x)dx which occurs in the original equation so.. indefinite ingegral ((u^-1)du) taking the antiderivative I get 1 + C is this right? thanks!
  46. J

    Integration do i use substitution or parts?

    Hi, i am stumped on how to start this: integration of sin(ROOT(1 + x)) so, basically sin of square root 1 + x thanks James
  47. G

    Direct integration by substitution

    Definite integration by substitution I just need a check on this, the book and I are getting different answers... The problem and my answer: http://www.mcschell.com/p14.gif http://www.mcschell.com/p14_worked.jpg The book gives 0.00448438 though. :confused: Thanks! -GeoMike-
  48. E

    Substitution Methods and Exact Equations

    There is a problem in my book which wants us to find the general solution to the given equation. I understand most of the problem it is just the integral part that is tricky. Here is the problem: x(x+y)y' = y(x-y) In this problem I know that you need to divide the equation by x and you...
  49. S

    What is the Substitution for Solving Second Order Differential Equations?

    I am having real trouble with this second order differential The substitution is given and i just can't seem to use it What am i missing here? x \frac{d^2 y} {dx^2} -2 \frac{dy} {dx} + x = 0, \frac{dy} {dx} = v All help welcome
  50. MathematicalPhysicist

    A question in substitution in logic.

    i need to prove that if x is the first of sub(\phi;a,\psi) then there exists 1<=i<=n and there exist firsts \phi' of \phi_i and \psi' for \psi such that x=sub(\phi';a,\psi)\psi' where sub(t;a,b) is defined as follows: let a1,..,an be n signs and b1,..,bn expressions. a=(a1,...,an)...
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