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. J

    How Does the Substitution Rule Work?

    Homework Statement If u=g(x) is a differentiable function whose range is an interval I and f is continuous on I, then Integral f(g(x)g'(x)dx=Integral f(u)du. Homework Equations The Attempt at a Solution
  2. L

    Seperable Equations using a substitution for a differential equation

    Homework Statement The d.e y' = (y+2x)/(y-2x) is NOT seperable, but if you use a substitution then you obtain a new d.e involving x and u, then the new d.e is seperable... Solve the original d.e by using this change of variable method Homework Equations I'm going to use the...
  3. D

    Substitution in double integral

    Homework Statement Use the transformation u = 3x + 2y and v = x + 4y to evaluate: The double integral of (3x^2 + 14xy + 8y^2) dx dy for the region R in the first quadrant bounded by the lines y = -(3/2)x + 1, y = -(3/2)x + 3, y = -(1/4)x, and y = (-1/4)x + 1. Homework Equations...
  4. W

    Solve Homogeneous DE Easier: Better Substitution?

    Given y' = y / (x + y^2), the substitution u = y^2 will give a homogeneous DE which can then be easily solved. Is there a substitution which would make things easier?
  5. R

    Integration using method of substitution

    Problem: (tdt)/(4-t^4)^(1/2) Attempt: I want the derivate of whatever i make u equal to, to equal something outside of u therefore I will factorize the denominator to equal -(-2+t^2)(2+t^2) and make u equal to (2+t^2) so that du=2tdt Balance the equation so that one side is equal to the...
  6. T

    Simplifying Integration by Substitution

    Homework Statement Using the substitution u² = 2x - 1, or otherwise, find the exact value of \int^{5}_{1} \frac{3x}{\sqrt{2x-1}}dx The Attempt at a Solution Right let's rearrange u in terms of x (i think that's how you say it): x = \frac{u^{2} - 1}{2}And now get an expression for dx u =...
  7. T

    Substitution Rule for Indefinite Integrals

    Homework Statement \int2e^-^7^xdx Homework Equations None The Attempt at a Solution (\frac{-2}{7})(\frac{e^-^7^x}{-7})+C This is as far as I can go, but the answer is: \frac{-2e^-^7^x}{7}+C
  8. R

    Integrate (x^3 + x^2)/(1 + x^4) using substitution?

    Integrate the following: (x^3+x^2)/(1+x^4) I have been taught only integration by substitution. My teacher told me that this can be solved using that ith some trick. I have tried for a long time. All that I can do was to convert the numerator to x^2(x+1) and the denominator to...
  9. R

    Possible webpage title: Solving Integrals Using the Substitution Method

    Evaluate the following integral using integration by substitution: http://img254.imageshack.us/img254/750/44900023cm4.png [/URL] Here is my attempt: Let x = sinu, then dx/du = cosu Substituting gives, ∫1/(1-sin2u)×cosu du = ∫1/(1-sin2u)×cosu du = ∫cosu/√cos2u du = ∫cosu/cosu...
  10. K

    Trig substitution integration problem, test in 1hr 30min

    Homework Statement √(9-x²) / (x²) Homework Equations Just trig substitution The Attempt at a Solution Ok, for trig sub I did u=asinΘ x=3sinΘ 9-x²=9-9sin²=9(1-sin²Θ) so putting it into the equation √9cos²Θ=3cosΘ/x^2 where do I go from here, I tried getting help at Math...
  11. Y

    Substitution for impulse in recurrence equation

    Hi, I was wondering what substitution to try when finding a particular solution for a recurrence equation with a linear combination of impulses on the right side of the equation. I think an example will clarify this. Given the recurrence equation y[k+2]-4y[k]=\delta [k] + 2\delta [k+1]...
  12. S

    A difficult substitution & separable integral

    Homework Statement I'm not sure how to proceed here. The first one asks me to find the area of a surface obtained by rotating the curve y = cos(x), 0 \leq x \leq\ \frac{\pi}{3} The second one asks to Solve: \frac{dy}{dt} = \frac{ty+3t}{t^2+1}\ y(2)=2 Homework Equations The Attempt at...
  13. N

    How Effective Is the Tan(x/2) Substitution for Solving ∫√(1-sin(x))dx?

    I'm looking to solve this integral with the tan(x/2) substitution but so far, I don't know what to do. Homework Statement \int\sqrt{1-sinx}dx Homework Equations u=tan(x/2) The Attempt at a Solution Well, using the tan(x/2) substitution, I get that: sinx=\frac{2u}{1+u^2}...
  14. N

    Integrate using trig substitution

    Homework Statement int sqrt 8x-x^2 Homework Equations trig sub The Attempt at a Solution complete the square integral becomes int sqrt 16-(x-4)^2 let x-4= 16sin(Q) sqrt 16-(x-4)^2 =sqrt 16-256sin^2(Q) dx= 16cos(Q)dQ =...
  15. D

    Integration using u substitution and arctan

    so I'm having problems with the coefficients in this problem. \int(10z+8/z^2-8z+41)dz i know that the main chunk is (a)ln|(z-4)^2+25|+(b)arctan((z-4)/5) a and b are 5 and 32/5 respectively the problem is i can't seem to split up the top so that the first portion is the derivitive...
  16. S

    Trigonometric Substitution for Integrating Radical Expressions

    Homework Statement Hey, it's me again. This method is giving me some trouble. This is the first problem: \int^3_0\ x^2\sqrt{9-x^2} \ dx The second problem is: \int\frac{dx}{\sqrt{2x^2+2x+5}}. How do I use a trig. substitution to start on this one? Homework Equations The Attempt at a...
  17. P

    Integral with trig substitution

    [SOLVED] Integral with trig substitution Homework Statement Find \int(x^3)/\sqrt{x^2-9} Homework Equations Trig substitution. sin^2 + cos^2 =1, and other things that you can figure out from that. Half angle formula, cos^2\theta=(1+cos(2\theta) )*.5 The Attempt at a Solution...
  18. S

    Substitution for indef. integral

    Homework Statement Hey folks, I think I know how to solve this by parts but I need a substitution to get there. I've been staring at examples for a while but I still don't understand how to apply the substitution rule. Anyway, here's the integral: \int x^9cos(x^5) Homework Equations...
  19. A

    Integration, trig with double substitution

    Homework Statement Integral ( cos(x)/(1+cos(x))^.5 dx) Homework Equations The Attempt at a Solution Integral ( cos(x)/(1+cos(x))^.5 dx) Square it Integral ( cos(x)^2/(1+cos(x) dx) Multiply by the Conjugate Integral ( cos(x)^2/(1+cos(x) * (1 - cos(x))/(1 - cos(x)) dx)...
  20. L

    Proving Continuity on R to Solving Question 5.5.64

    http://www.math.cmu.edu/~handron/21_122/hw/hw1.pdf My question is about number 5.5.64 on that page. "If f is continuous on R, prove that ..." I have been unable to do this and would appreciate help.
  21. W

    Which Compound Undergoes Unimolecular Substitution Fastest?

    Homework Statement I have these compounds, and I have to predict which one will undergo unimolecular substitution reaction the fastest. And which one will undergo it the slowest. 3 bromo cyclohexene 1 bromo cyclohexene 4 bromo cyclohexene 2 bromo hexane (not cyclohexane/ene) The...
  22. ~christina~

    Is Direct Substitution Effective for Solving Linear Wave Equations?

    Homework Statement verify by direct substitution that the wave function for a standing wave given in equation below is a solution to the general linear wave equatin. y= (2A sin kx)cos \omega t \frac{\delta^2y} {\delta x^2}= \frac{1} {v^2} \frac{\delta^2 y} {\delta t^2} Homework...
  23. O

    Integration with Trig Substitution

    Homework Statement Ok, so I was doing a problem on the electric field strength of a continuous charge distribution and I arrived at this seemingly easy integral \int \frac{1}{({l^2+a^2})^\frac{3}{2}} dl sorry the latex is lagging badly, you can see the correct integral by clicking on it. it...
  24. I

    How Do You Calculate the Interest Rate for Compound Interest Problems?

    Have this question in relation to some investment exam I am doing, I am a maths novice being some years since leaving school etc, ok enough of the excuses. Example FV = future value PV = present value R = interest rate N = number of compounding periods my PV is 6000 and my FV is...
  25. J

    Integration with Trigonometric Substitution

    [SOLVED] Integration with Trigonometric Substitution Homework Statement Given integral (I): I[(x)sqrt(9-x^2)dx] by words: Integral of "X" times square root of "9-X(squared) Use proper trigonometric substitution to solve this problem. Homework Equations The Attempt at a Solution
  26. I

    Evaluating integrals by trigonometric substitution

    I have a few quick problems concering evaluating integrals by trigonometric substitution. I guess I will just post five that way if anyone can help with any, would be greatly appreciated. Also: if anyone could inform me on how to input the actual equations onto this forum as I have seen in some...
  27. Simfish

    Equivalence of Irreducibility Tests for Polynomials Under Variable Substitution

    So if we have, say, a polynomial f(x) = a_n x^n + ... + a_0 and, say, use the substitution x = y + a, then do ALL irreducibility tests work the same? And do all OTHER tests also work the same? Is the polynomial FUNDAMENTALLY the same? And what theorem is there to prove this?
  28. J

    Challenged by my teacher: integration by substitution

    I just learned how to integrate through substitution and I was challenged by my teacher with an apparently easy problem but I'm really struggling with it. He said he will give me an F if I don't solve it for tomorrow, I guess this is what I get by being the one who always understand in class...
  29. 2

    Calculating the Density of a BCC Crystal with Ta Substitution

    how would i start this solution off? we replace 7.5 atomic % of the chromium atoms in its BCC crystal with tantalum. X ray diffraction shows that the lattice parameter is 0.29158 nm find the density.
  30. Saladsamurai

    Differential Eq. (Substitution)

    Homework Statement Solve by making an appropriate substitution. I am given the homogeneous DE:xdx+(y-2x)dy=0 Now we have bee using either y=ux or x=vy. . . I tried both, but the latter seemed easier. x\frac{dx}{dy}+y-2x=0 letting x=vy and dx/dy=v+y*dy/dv vy(v+y\frac{dy}{dv})+y-2vy=0...
  31. S

    Free fall far away from Earth (integral substitution problem)

    [SOLVED] Free fall far away from Earth (integral substitution problem) Homework Statement Given: v(x) = -v_1\sqrt{\left(\frac{R}{x} - \frac{R}{h}\right)} Find the time t. Homework Equations Listed above where v_1 , R , h are all constant. The Attempt at a Solution v(x) =...
  32. M

    Integrating cos^-2(x) after inverse substitution

    Homework Statement Use the substitution x=4sin(t) to evaluate the integral: S 1/[(16-x^2)^(3/2)] dxHomework Equations x = 4sin(t) The Attempt at a Solution x = 4sin(t) dx = 4cos(t) dt 4cos(t) = (16-x^2)^(1/2), i cube both sides to get (4cos(t))^3 = (16-x^2)^(3/2), then plug in dx and...
  33. I

    Use substitution to evaluate the integral

    Homework Statement Use substitution to evaluate the integral Homework Equations ∫√(cotx) csc^(2)x dx?
  34. B

    Integration using Trig Substitution

    Homework Statement \int \frac{cosx dx}{\sqrt{1 + sin^{2}x}} Homework Equations Expression: \sqrt{a^{2} + x^{2}} Substitution: x = a*tan\Theta Identity: 1 + tan^{2}\Theta = sec^{2}\Theta The Attempt at a Solution I have tried using Trig Substitution, but I end up getting an equation much...
  35. N

    Trig Substitution for Integrating sin^3(x)/cos(x) - Homework Help

    Homework Statement \int ((sin(x))^3/(cos(x)) )*dx The Attempt at a Solution alright i have been trying to use u= cosx -du = sinx but it doesn't make sense bause there is still a sinx^2 to account for so i know i need to make a trig substitution but i can't figure out the appropriate...
  36. Q

    Trig Substitution: Solving for the Missing Identity

    [SOLVED] More trig substitution help... I've looked at this problem about 3 times and still can't figure it out...where identity did they use to substitute out the part in the red box? Thanks for the help
  37. A

    Simple Integration using U Substitution

    Homework Statement Find the indefinite integral. The antiderivative or the integral of (x^2-1)/(x^2-1)^(1/2)dx Homework Equations The Attempt at a Solution Tried using (x^2-1)^(1/2) as u and udu for dx and I solved for x but I am still left with a 1 on top not sure how to...
  38. A

    How can I integrate this function using substitution?

    Homework Statement Evaluate the definate integral of the following \int (from 1 to 2) \frac{sin t}{t} dt The Attempt at a Solution I am actually stuch from the very beginning. I tried to set u=sin(t) but this doesn't help much because (sint)'=cost and this is going to make the...
  39. M

    Need help: integration by substitution.

    Hello all, how are you? we are currently working on integration by substitution, what do you guys think about the way i solved this one: Find: \int \frac{(t+1)^2}{t^2} dt My solution: \int \frac{(t+1)^2}{t^2} dt = \int 1dt + \int \frac{2}{t} dt + \int \frac{1}{t^2} dt = t +...
  40. G

    Substitution with double integral

    Homework Statement Using transforms: u = 3x + 2y and v = x+4y solve: \iint_\textrm{R}(3x^2 + 14xy +8y^2)\,dx\,dy For the region R in the first quadrant bounded by the lines: y = -(3/2)x +1 y = -(3/2)x +3 y = -(1/2)x y = -(1/2)x +1 I'm itching to see where I've gone wrong on this one...
  41. C

    Integration by substitution for (1+x)/(1-x)

    Homework Statement I want to integrate (1+x)/(1-x) Homework Equations The Attempt at a Solution I have looked at many examples of substitution method - this one appears simple but I am not finishing the last step... - I know you must first take u=(1-x) - Then du = -dx what...
  42. P

    How Does Substitution Simplify the Integration of \(3x^2(x^3 - 2)^4 dx\)?

    1. Find, by substitution, the integral of; 3x2(x3 - 2)4 dx 2. susbt' 3. u = x3 - 2, so du/dx = 3x2, and du = 3x2 dx Now this is where I'm not sure what to do. As u = x3 - 2 you know that x = (u + 3)1/3, and so i think you can write the integral as; \int(u+3)1/3.u4 du ... but i when i look...
  43. H

    Integrating Trigonometric Functions with Substitution: A Quick Guide

    Homework Statement \int1/[Sin[x]\sqrt{}((Sin[x])^2+k)] The Attempt at a Solution I don't have any idea of the solution. Mathematica gives the answer as -(1/sqrt(k))ArcTanh[(Sqrt(2k)Cos(x))/sqrt(1+2k-cos(2x)]
  44. U

    Integrating (x^2+1)^{-3/2} using Substitution

    Hi all, I've been studying calculus out of Tom Apostol's book "Calculus". I'm having troube with the following problem in the section on integration by substitution: Integrate \int(x^2+1)^{-3/2}\,dx. I tried the substitution u=x^2+1 but it didn't seem to work. I can't see anything else...
  45. L

    Integration by u- substitution (involving natural logs)

    Homework Statement \int \frac{1}{1+\sqrt{2x}}dx Homework Equations u=1+\sqrt{2x} \sqrt{2x}=u-1 dx=(u-1)du The Attempt at a Solution I was able to get it down to: \int (1-\frac{1}{u})du = u-\ln{lul}}+C = 1+\sqrt{2x}-\ln{l1+\sqrt{2x}l}+C However, my book says that...
  46. F

    Where Did I Go Wrong with This Trig Substitution for x^3/sqrt(1-x^2)?

    For the integral \int frac{x^3}{sqrt{1-x^2}} dx} ==> okay... what I meant was: int of x^3 over sqrt(1-x^2) --I trig substitute to get sin^(3)(x)cosxdx over cos x and end up with sin^3(x)...this is obviously wrong, can anyone point out what i did wrong?
  47. R

    Solving Substitution Process Homework Questions

    Homework Statement 1) antiderivative of ((t^2)+2)/((t^3)+6t+3) dt 2) antiderivative of r(sqrt((r^2)+2))dr help please with these Homework Equations The Attempt at a Solution #2 let u = r^2 + 2 du/dr = 2r du = 2rdr?? i don't knoww!
  48. M

    Integrating substitution problem?

    [SOLVED] Integrating substitution problem? Homework Statement Sorry to hijack this thread sort of (as a similar named one already exists), but the title is aptly suited to my question. I have integral to integrete and I don't really know how to do it tbh. . . s=\int{\sqrt{2+(3t)^2}dt...
  49. S

    Integration using substitution

    [SOLVED] Integration using substitution Problem: Find the integral of: \int\sin^{6}\theta\cos\theta d\theta My attempt: Let u\equiv\cos\theta so: du\equiv\sin\thetad\theta Only I don't know where to go from there. The book says it should \frac{1}{7}\sin^{7}\theta+C but I have no idea how...
  50. P

    Integration by Substitution fails

    Question: \int^{1}_{-1} \frac{dx}{(1+x^4)} I attempt: u = x^2, so x= u^1/2 dx= 1/2 u^(-1/2) Which gives me \int^{1}_{1} \frac{1}{(1+x^4)} * \frac{1}{(2u^1/2)}du, which is 0. Thats not the answer as seen by any graphing utility. Where is this error? I do not know integration by parts. I just...
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