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. S

    How can substitution be used to solve a differential equation?

    Homework Statement The problem states: Use substitution to solve: y'=1/(x+y)^2-1Homework Equations The Attempt at a Solution I used the substitution of v=x+y resulting in the answer y=[3(x-C)]^(1/3) - x but I'm not too sure that's right Can some help with the answer and the steps for...
  2. P

    Schrodinger equation reduction using substitution

    SOLVED: Schrodinger equation reduction using substitution Homework Statement Given \frac{d^2 \psi}{dx^2} - Ax\psi + B\psi = 0 make a substitution using w= A^{1/3} (x - \frac{B}{A}) to get \frac{d^2 \psi}{dw^2} - w\psi = 0 Homework Equations The Attempt at a Solution...
  3. S

    Line integral, problems with substitution (probably)

    Homework Statement I am trying to solve a line integral (bear with me, I am new to calculus!) and my basic skills of integration seem to fail me. I am sure the mistake is quite obvious, as I keep getting the wrong answer, 2, when it should be ~2.69 Homework Equations \int_C 4x^3dS C is the...
  4. B

    Difficult integral for Trig Substitution

    ok i have been studying the in-depth processes of trigonometric substitution with integrals and this problem has me frusterated. \int x^2\sqrt{(x^2-4)} dx The evaluation is clear (from an old Table of Integrals I found), but the derivation is not at all clear, which is what i want to know. I...
  5. J

    Inverse Substitution: Solving for z in Terms of x

    Homework Statement Let's say you make the inverse substitution x = 2tan(z) in some integral. Let's say you evaluate the integral get something like like 4sin(z). How do you put z back in terms of x? Homework Equations The Attempt at a Solution I can do it by saying arctan(x/2) =...
  6. K

    Integration using Hyperbolic Trig substitution

    Homework Statement Evaluate: \int\\{1}/{\sqrt{x^2-1}} dx between -3, -2 I know I'm supposed to use hyperbolic substitution in the question. Homework Equations edit: cosh^2(t) - sinh^2(t) = 1 The Attempt at a Solution let x = -cosht, inside the integral let dx = sinh(t) dt int (...
  7. N

    How do you simplify the integral after trigonometric substitution?

    I’ve got this tricky trig substitution that I’ve been trying to do as for my Calc II class. I think that I’ve done the substitution part right I just have no idea where to go from here. I’ve tried many routes including integrating by parts as well. Please help me thank you...
  8. A

    Integration by substitution for integral

    Homework Statement Use substitution to evaluate the integral. \int \frac{4cos(t)}{(2+sin(t))^2}dt Homework Equations None, really.The Attempt at a Solution I'm not sure what to use as u, for the substitution. I've tried (2+sin(t))^2, as well as other attempts, but I can't seem to find anything.
  9. mvantuyl

    Integration by trig substitution

    Homework Statement Integrate: \int\sqrt{1 - 9t^{2}}dt Homework Equations The Attempt at a Solution t = 1/3 sin x dt/dx = 1/3 cos x dt = 1/3 cos x dx 3t = sin x 1/3 \int\sqrt{1 - sin^{2} x} cos x dx 1/3 \int cos ^{2}x dx 1/3 \int(1 + cos 2x) / 2 dx 1/6 \int1 + cos 2x dx...
  10. mvantuyl

    Integration by trig substitution

    Homework Statement Integrate: \int\sqrt{1-9t^{2}}dt Homework Equations The Attempt at a Solution t = 1/3 sin\Theta dt/d\Theta = 1/3 cos\Theta dt = 1/3 cos\Thetad\Theta 3t = sin\Theta 1/3\int\sqrt{1-sin^{2}}\Theta cos\Thetad\Theta 1/3\int cos^{2}\Thetad\Theta 1/3\int(1 +...
  11. J

    Can someone please check my answer: Trig Substitution problem

    I tried to use the latex reference but it was way too complicated for me. Can you see the attachment. Thank you!
  12. K

    Integration by parts and substitution help

    Integration by parts and substitution help! Homework Statement ∫0 -1 e ^√x+1 Homework Equations The Attempt at a Solution
  13. M

    Finding the Number of Simple Substitution Ciphers with No Fixed Points

    Homework Statement Question: How many simple substitution ciphers are there where no point is fixed (ie: no letter is mapped to itself)? EDIT: Incase the termonoligy is different, a simple substitution cipher is a mapping where the plaintext in english is encoded so that every letter is...
  14. JJBladester

    Differential Equation - Solve by substitution

    Homework Statement Solve the following differential equation by using substitution. Homework Equations (x + 2y - 1)dx + (2x + 4y - 3)dy = 0 Let u = x + 2y The Attempt at a Solution u = x + 2y du/dx = 1 + 2(dy/dx) (x + 2y - 1)dx + (2x + 4y - 3)dy = 0 (u - 1)dx + (2u -...
  15. W

    Ordinary Differential Equation by substitution.

    Homework Statement dy/dx = (x+3y)/(3x+y) I have to solve the given differential equation by using an appropriate substitution... The Attempt at a Solution I used algebra to make the equation (3x+y)dy - (x+3y)dx = 0 then made x = vy and dx = vdy + ydv. then plugged into get...
  16. W

    Finding a limit using substitution rule, my answer is 0, my book`s is -2

    Homework Statement Lim [(tanx)^2] / [1 + secx] <<< as x goes to pi Homework Equations The Attempt at a Solution (tan x)^2 = (sin x)^2 / (cos x)^2 (sin x)^2 = y lim y = 0 <<< as x goes to pi lim [y/ (cos x)^2] / [1 + (1/y)] <<< as y...
  17. K

    Trigonometric Substitution Problem - Calculus 2

    Hey there This is a trig substitution for my Calculus 2 class and I really have NO idea how to get started... \int\frac{4}{\sqrt{3-2x^2}}dx My professor has yet to go over how to evaluate trigonometric substitutions with coefficients in front of variables.
  18. R

    Substitution in Differential Equations

    Make an appropriate substitution to find a solution of the equation dy/dx=sin(x-y). Does this general solution contain the linear solution y(x)=x-pi/2 that is readily verified by substitution in the differential equation? Here's what I did: v=x-y y=x-v y'=1-dv/dx 1-dv/dx=sin(v)...
  19. tony873004

    How do I start this ODE substitution problem?

    Homework Statement \left( {x + y} \right)y' = x - y Homework Equations Back of book: x^2 - 2xy - y^2 = C The Attempt at a Solution I'm not sure how to start this problem. In the examples in the book, they make a substitution, v=something, and all that was left were v's and...
  20. F

    Integrating v^2/v^2+4 - Exploring Arctan & u Substitution

    Homework Statement how do i integrate v^2 / v^2 + 4 Homework Equations i understand this has something to do with arctan but if i use u substitution to let v=(u/2) so (on the bottom) it becomes (1/4)(1+(v/2)^2) there's still a v^2 on the top which the u substitution does not...
  21. L

    Integration By Substitution It's Been A While

    Homework Statement It's been god knows how long since I've had to use integration by substitution. I've totally forgotten it. I am trying to integrate to solve for the value of an electric field at a given point. The integral I am trying to solve is...
  22. G

    Direct substitution in the analysis of periodic motion

    Homework Statement By direct substitution, show that equation (3) is a solution of the differential equation (2). Homework Equations (2) (d^2 θ)/(dt^2 )=-g/l θ (Second derivative of θ(t)=-g/l θ.) (3) θ(t)=θ_0 cos⁡(√(g/l) t) The Attempt at a Solution I...
  23. A

    Calculus Substitution Rule Problem Check

    Homework Statement Evaluate the indefinite integral... \int x^2 (x^3+5)^9 dx Homework Equations \int f(g(x))g'(x)dx = \int f(u)du The Attempt at a Solution u = x^3+5 du = x^2dx So my answer is... Does that look right? And one more... Homework Statement Evaluate the indefinite...
  24. C

    Use trigonometric substitution to verify

    Homework Statement Using trigonometric substitution, verify that \int \sqrt{a^2[-t^2}dt (INTEGRAL FROM 0 TO \pi)=(1/2)a^2sin^{-1}(x/a)+(1/2)x\sqrt{a^2-x^2 Sorry it doesn't seem to want to allow me to place superscript inside a square root. but inside the first sq. root is a2-t2 and the...
  25. C

    A trigonometric substitution problem

    Homework Statement \intsin3cos2xdx The Attempt at a Solution I've successfully solved this problem by factoring out 1 sinx and changing the sin2x to (1-cos2x then assigning u=cosx and du=-sinx and so on. What I'm wondering is why does letting u=sin3x in the original integral not...
  26. B

    Help nedded to solve a D.E. using substitution

    Homework Statement Use the appropriate substitution to solve the following D.E.: -ydx + (x + \sqrt{}xy)dy = 0 Homework Equations y = ux The Attempt at a Solution y = ux implies dy = udx + xdu so -xudx + (x + x\sqrt{}u)(udx + xdu) =0 we then get after some simplificaion...
  27. V

    Prove two integrals are the same using U substitution

    Homework Statement If a and b are positive numbers, show that \int_0^1 x^a*(1-x)^b\,dx = \int_0^1 x^b*(1-x)^a\,dx using only U substitution.Homework Equations Just U substitution and the given equation--I can't use multiplication rules or anything like that; otherwise it would be easy.The...
  28. Z

    Equation w/ Homogeneous Coefficients - y=ux substitution

    Differential Equation w/ Homogeneous Coefficients - y=ux substitution I am teaching myself, this problem is from ODEs by Tenenbaum and Pollard. This is not homework for a class. Homework Statement (x+y){dx} - (x-y){dy} = 0 Homework Equations y=ux, {dy} = u{dx} + x{du} The...
  29. P

    Transforming Limits of Integration for Variable Substitution

    Homework Statement \int\int\int _E\(x^2y}\;dV Where E is the solid bounded by x^2/a^2+y^2/b^2+z^2/c^2=1 Homework Equations variable substitution x=au, y=bv, z=cw. The Attempt at a Solution I found the jacobian (abc) and I substituted my variables but I can't find the limits of...
  30. P

    I need to find Lim (x->0) arcsin(2x)/arcsin(3x) by substitution

    I need to find Lim (x->0) arcsin(2x)/arcsin(3x) I can do a substitution arcsin(2x) = y => 2x = sin(y) and get arcsin(sin(y)) for the nominator, which is equal to y. However, for the denominator i get arcsin(3/2 sin(y)) which I'm not sure what to do with. Am I on the right path?
  31. R

    Integrating Trigonometric Functions with Evasive Substitutions

    I am staring at an integral of the form \int \frac{sin(at)}{(1 + bsin^{2}(at))^{1/2}} dt which I have generated for myself (in attempting to model the behaviour of a particle in an oscillating field). I can't see a sensible substitution to try, at present. I could hunt down a standard...
  32. A

    Is this allowed in substitution

    Homework Statement I have recurrence relation problem and what to ask would my way be just as correct as the TA did the solution: f(n) = 0, n=1 and 2f(n/2) + n -1 The Attempt at a Solution Here we assume n = 2k This is my way: f(k-1) = 2[2f(2k-2)*2k-1-1] + 2k - 1 =...
  33. A

    How Does Repeated Substitution Solve the Recurrence Relation f(n)?

    Homework Statement f(n) = 0 when n=1 and 2f(n/2) + n - 1 when n is otherwise Homework Equations Repeated substitution The Attempt at a Solution f(n) = 2f(n/2) + n - 1 = 22f(n/22) + n - 3 = 23f(n/23) + n - 7 = 24f(n/24) + n - 15 Guess: f(n) = 2kf(n/2k) + n...
  34. H

    How can trigonometric substitution be used to simplify a complex integral?

    Hello, I have been wokring on this problem for some hours now, and I get the wrong answer, but I can't understand why, could you guys please look at it? http://img219.imageshack.us/my.php?image=utennavnwv7.jpg
  35. C

    Integration by trigonometric substitution

    Homework Statement Solve the integral using trigonometric substitution. \int\frac{\sqrt{4x^{2}+9}}{x^{4}}dx Homework Equations 2x=3tan\theta x=2/3 tan\theta dx=2/3 sec2\thetad\theta \sqrt{4x^{2}+9}=3sec\theta The Attempt at a Solution \frac{8}{9} \int...
  36. G

    Why Is the Slope of the Indifference Curve Always Negative?

    I understood the concept and definition of marginal rate of substitution , and also MRS = -(dy/dx) , but why is dy/dx never positive ?
  37. P

    Is My Trigonometric Substitution Correct for This Integral?

    Homework Statement \int \frac{e^{3x}dx}{\sqrt{1-e^{2x}}} Homework Equations The Attempt at a Solution Alright so I am able to do other similar problems fine, I think it is the "e" that is throwing me off as well as the fact that the "x" is in the exponent. I started the...
  38. G

    Capacitor substitution for TEA nitrogen laser

    My next project is to build a simple TEA nitrogen laser, as seen here: http://photonics.tfp.uni-karlsruhe.de/1/a-homemade-uv-laser.html Would it be acceptable to substitute two strontium titanate ceramic caps, each rated 50KVDC @ 910pf, in place of the suggested aluminum foil / plastic...
  39. J

    Proving a differential equation using the substitution method

    Homework Statement dy/dx = \frac{[2cos(x)^2-sin(x)^2+y^2]}{[2cos(x)]} Substitute y(x) = sin(x) + \frac{1}{u(x)} Homework Equations By doing the substitution, it will yield the differential equation for u(x) du/dx = -u tan(x) - \frac{1}{2}sec(x) The Attempt at a Solution I figured out i...
  40. R

    Linear Differential Equation with a substitution.

    Wow I am so very bad at differential equations.. :( The problem Here is the exact problem I'm given: http://img229.imageshack.us/img229/9025/10640260bs5.jpg attempt at a solution I'm guessing that I need to differentiate y(x) that I am given and substitute that into the left hand side and...
  41. D

    Trig Substitution Homework: Solving \int{cos^4 6x sin^3 6x dx}

    Homework Statement \int{cos^4 6x sin^3 6x dx} Homework Equations The Attempt at a Solution I've gotten this far but now I'm stuck: \int{cos^4 6x sin 6x sin^2 6x dx} = \int{cos^4 6x}*(\frac{1-cos 12x}{2})sin 6x dx
  42. T

    Trigonometric Integration and U Substitution

    Hi, We were going over trigonometric integration in Calculus II the other day. I got the basic idea, but get lost when we're doing the u-substitution. We had a problem like this: \int cos^3 (x) dx Then we did: \int (1 - sin^2 (x)) cos(x) dx Starting u-substitution: u =...
  43. S

    Integrate exp(-(x^2)) using the substitution u=tanh(x)

    The question is to convert the infinity limits of the integral \int^\infty_{-\infty} e^{{-x}^2} dx to finite limits \int^{u_a}_{u_b} g(u) du using the substitution u = tanh(x). How do I go about it?
  44. J

    V substitution in homogeneous equations (diff eq)

    Hey all, i think I'm doing most of this right, but I'm missing a coefficient somewhere when integrating or something... Homework Statement Substitute v=y/x into the following differential equation to show that it is homogeneous, and then solve the differential equation...
  45. Y

    Solving f(x) Substitution: 1/x +1-(1/x)/h

    Homework Statement f(h+1)-f(h)/h. If f(x)=1/x, simplify. Homework Equations The Attempt at a Solution 1/x+1-(1/x)/h is the answer 1/h? I am not sure if i substituted this correctly or if I solved this right. I put 1/x in everywhere there was a f(h).
  46. K

    Long division with zero to nine substitution

    Substitute the letters by a different digit from 0 to 9 to satisfy this cryptarithmic long division problem. ....N K T ...------------------- F A R |...F R M N K B ...A K K N ...--------------------- ..... A B I K ...AM O K ...--------------- .....R T I B .....R A K T...
  47. P

    Integral, Trig Substitution - Stuck

    Homework Statement Use trigonometric substitution to evaluate \int{\frac{x^2}{\sqrt{9-x^2}}}dx The Attempt at a Solution Let x=3\sin\theta then dx=3\cos\theta d\theta \int{\frac{x^2}{\sqrt{9-x^2}}}dx =\int{\frac{9\sin^2\theta}{3\sqrt{1-\sin^2\theta}}}\ 3\cos\theta \ d\theta...
  48. A

    Help with using Integration By Substitution

    1. Homework Statement : using the substitution u=3x+4, work out: \int 2x \sqrt{3x+4}2. The attempt at a solution: \int 2x \sqrt{3x+4} u=3x+4 \rightarrow \mathrm{d}x = \frac{\mathrm{d}u}{3} \int \left( \frac{u-4}{3} \right)\left(u^{\frac12}\right) \ \frac{\mathrm{d}u}{3} \frac19 \int u^{\frac32}...
  49. S

    1-dimensional substitution rule for integrals

    This is kind of an awkward post, but: The only topics that have always bugged me in calculus I-III are those which deal with differentials... I've convinced myself (and proven) that if a function can be written as f(x)dx = g(y)dy, then it is possible to "separate" the variables and solve the...
  50. L

    Trigonometric Substitution Proof

    The question is: Use x = \tan \theta , \frac{-\pi}{2} < \theta < \frac{\pi}{2} to show that: \int_{0}^{1} \frac{x^3}{\sqrt{x^2+1}} dx =\int_{0}^{\frac{\pi}{4}} \tan^3 \theta \sec \theta d\theta Using that substitution, I got it down to: \int_{0}^{\frac{\pi}{4}} \frac{\tan^3...
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