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. karush

    MHB What is the Integral of Square Root of T over T+1 with Substitution?

    $\displaystyle \int_0^4 {\frac{\sqrt{t}}{t+1}}dt $ $\displaystyle U=\sqrt{t}\ \ \ t=U^2 \ \ \ dt=2Udu $ $\displaystyle \frac{\sqrt{t}}{t+1} \Rightarrow \frac{U}{U^2+1} $ $\displaystyle \int_0^4 \frac{U}{U^2+1} 2Udu $ if ok so far tried $U= sec^2(\theta)$ but couldn't not get answer which is
  2. MarkFL

    MHB Integral of sqrt(1+x^2)/x: Help Solving w/ Trig Substitution

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

    MHB How Can I Integrate x*sqrt(1-x^4) Using Trig Substitution?

    how do i go about integrating x*sqrt(1-x^4)?? i have no idea
  4. A

    MHB Integrate sqrt(1 + x^2) / x: Trig Substitution

    how do u integrate sqrt(1 + x^2) / x?? i reduced this to sec^3(u)/tan(u) but how do i go from here??
  5. T

    MHB Solving u Substitution Problem: Tim's Question

    Hi, I'm working on a u substitution problem so that. u = 3-x so that du = (-1) dx , or (-1) du = dx . With these equations you just switch content from one side to the other with no problems? Thanks, Tim
  6. P

    U substitution. Why -1/x^2 is my du?

    Homework Statement ∫(1/x^(2))(3+1/x)^(3) Homework Equations U substitution is the way to go here The Attempt at a Solution My problem is that I can't figure my du and what is next. I know which one it is but I don't know the reason for it. u=3+1/x du= I chose ln|x| first but...
  7. karush

    MHB Integral solve by using trig. substitution

    Evaluate integral by using $x=3\sin{\theta}$ $\int{x^3\sqrt{9-x^2}}\ dx$ substituting $\int{27\sin^3{\theta}}\sqrt{9-9\sin^2{\theta}}\Rightarrow 81\int\sin^3\theta\cos\theta\ dx$ since the power of sine is odd then $81\int\sin^2{\theta}\cos{\theta}\sin{\theta}\ dx$...
  8. karush

    MHB Solve trig integral by substitution

    this is supposed to be solved with U substitution $\displaystyle \int \sin^6(x)\cos^3(x) $ since \cos^3(x) has an odd power then $\displaystyle \int \sin^6(x)\left(1-\sin^2(x)\right)\cos(x) dx $ then substitute u=\sin(x) and du=cos(x)dx $ \int u^6 \left(1-u^2\right) du $ so if ok so far
  9. Radarithm

    Integrating by Substitution: Evaluating \int \frac{3x}{x^2+2}

    Homework Statement Evaluate: \int \frac{3x}{x^2+2} Homework Equations \int \frac{1}{u} \frac{du}{dx} dx = \ln u + C The Attempt at a Solution I got a horribly wrong answer: \frac{1}{2x}\ln (x^2+2)+C This was done by multiplying \frac{du}{dx} by \frac{3x}{u} This part is what...
  10. S

    Integrating with substitution methods (part 1)

    Mod note: Edited the LaTeX so that the exponents show up correctly. Homework Statement This is from my Calculus II exam practice papers. We're currently dealing with different substitution methods (whichever apply to the given problem).Homework Equations \int \frac {\sqrt{1 - x^2}} {x^{4}}...
  11. MarkFL

    MHB Integral of [1/sqrt(8x-x^2)] dx - Daisy Dee's Answer at Yahoo Answers

    Here is the question: I have posted a link there to this thread so the OP can see my work.
  12. S

    MHB Evaluating definite integrals via substitution.

    Can someone make sure I'm on the right track with this problem? I'm a little confused because I thought that when you make a substitution you update the limits and get better numbers to work with when you plug them in the function in the end...Yet, it seems like I almost got worse numbers to...
  13. A

    Higher Order Differential Equation: Substitution

    Homework Statement Solve x^{2}\times y'' - 4 \times x \times y' + 6 \times y = 0 for y(x) by first using the substitution v = ln(x) to obtain an equation involving y, dy/dv, d^2y/dv^2 and no x. Solve for y(v), then return to y(x). Homework Equations NA The Attempt at a Solution I know how...
  14. A

    Rational functions and link with direct substitution property

    Homework Statement Hello, I know the direct substitution property in calculus. But, the definition of a rational function still confuses me. For example, are these rational functions: y=(x^2+2x+1)/(x+1) y=((x^2+2)^(1/2))/(x+1) The denominator of the first one could cancel. So...
  15. M

    How Does Dirac Delta Substitution Relate to Helmholtz's Decomposition Theorem?

    Hi All, I found (Wikipedia page on Helmotz's decomposition theorem) the follwoing equality, which puzzles me: $$\delta(x-y) = - (4 \pi)^{-1} \nabla^{2} \frac{1}{\vert x - y \vert}$$ I am not sure I understand, the r.h.s seems to me a proper function. The page mentions this a sa position...
  16. W

    Fierz Identity Substitution Into QED Lagrangian

    Hi all, I've been playing around with spin 1/2 Lagrangians, and found the very interesting Fierz identities. In particular for the S x S product, (\bar{\chi}\psi)(\bar{\psi}\chi)=\frac{1}{4}(\bar{\chi} \chi)(\bar{\psi} \psi)+\frac{1}{4}(\bar{\chi}\gamma^{\mu}\chi)(\bar{\psi}\gamma_{\mu}...
  17. G

    Trig Substitution for ∫x^3/(4x^2+9)^(3/2): Solving the Insanity

    ∫x3 ---------------------- (4x2 + 9) 3/2 According to my book this is a trig substitution integral. The normal procedure is to substitute atanθ for x when one has a square root w an argument of the form x^2 + a^2. Because the argument of the square root is 4x2 + 9, as opposed to simply x2...
  18. T

    Trig Substitution: Solving Homework Equations

    Homework Statement Homework Equations The Attempt at a Solution This isn't really a traditional question, but can someone explain to me how substituting u = tan^-1(x/y) got to that final value? I'm trying to understand this for an exam coming up.
  19. Feodalherren

    Trig Sub Homework: Did I Make a Mistake?

    Homework Statement Did I make a mistake here somewhere? The solution in the back of the book is completely different. Seems like they used trig sub one step later or something. I can't find any error in my logic. Test coming up soon and I'm confused and panicking -_-! Actually I just found a...
  20. G

    Substitution of imaginary variables in integral?

    I wanted to do this integral $$\int_a^b \frac{dx}{1-x^2} $$ and I was able to get the right answer with the substitution u=ix, where i is the square root of -1. But is this a valid mathematical procedure? $$\int_a^b \frac{dx}{1-x^2}=i \int_{-ia}^{-ib} \frac{du}{1+u^2}$$ Do those limits...
  21. F

    Trigonometric Substitution problem

    1. ∫\frac{\sqrt{x^{2}-4}}{x} dx, x=2secθ, dx=2secθtanθ dθ 2. \sqrt{x^{2}-a^{2}},sec^{2}θ-1=tan^{2}θ 3...
  22. F

    Theta ranges for trig substitution

    My professor, when doing trig substitution in lecture, always defines theta between certain intervals and when he takes the square root, he adds an absolute value bar to the trig function and then makes sure its positive through the interval. For practical purposes, is it necessary to go through...
  23. M

    MHB Can you spot the error in my substitution method for solving this DE?

    Hello all, it's been a long time. Hoping I can get some assistance with what is probably a simple substitution problem, yet it's flummoxing me. $$\frac{dy}{dx} = \frac{y+t}{t}$$ I've tried substituting $$ v = y+t $$ $$ y = v - t $$ $$ \frac{dy}{dx} = \frac{dv}{dt} - 1 $$ $$\frac{dv}{dt}-1 =...
  24. E

    MHB Roots of polynomial equations ( Substitution )

    How do I reduce u^4 + 5u^3 + 6u^2 + 5u + 1 = 0 to v^2 + 5v + 4 = 0 by using v = u + 1/u ?
  25. P

    Cylindrical continuity eq using cartesian substitution.

    I've have two questions, but if my assumption is incorrect for the first, it will also be incorrect for the second. (in-terms of physics.) For a two dimensional cylinder, using cylindrical co-ordinates (as follows), taking v(subscript-r) => velocity normal to cylinder surface & v(subscript-phi)...
  26. Y

    MHB Integral - substitution method problem

    Hello all I am working on this integral \[\int \frac{x^{2}+1}{x^{4}+1}dx\]Now, I have tried this way: \[u=x^{2}+1\] after I did: \[\int \frac{x^{2}+1}{\left ( x^{2}+1 \right )\left ( x^{2}-1 \right )}dx\] But I got stuck, I got: \[\frac{1}{2}\cdot \int \frac{1}{u\sqrt{u-1}}dx\] I thought...
  27. Y

    MHB Integral with substitution method

    Hello I need to solve \[\int \sqrt{x^{3}+4}\cdot x^{5}dx\] using the substitution method. I did \[u=x^{3}+4\] but I got stuck with it. thanks!
  28. Mathelogician

    MHB Question on substitution of variables in natural deduction of predicate logic.

    Hi all, I need Explanation on the attached image from Van Dalen's Logic and Structure; specially on how the red part follows from the lines before it! Regards.
  29. Y

    MHB Solving Integrals with Substitution

    Hello, I need some help solving this integral, \[\int \frac{\sqrt{x}}{\sqrt{x}+1}dx...u=\sqrt{x}+1\][\tex] After I make the substitution I get stuck a little bit, because I can't get rid of dx. and also this one, same principle, \[\int x^{3}\cdot \sqrt{7+3x}\cdot dx ...u=7+3x\] how do I...
  30. BruceW

    Limits of integral after substitution

    Hi everyone, I am having a 'crisis of faith' in how the limits of an integral should change when you make a substitution for the variable involved. Especially when using a sinusoid substitution, since the sinusoidal functions are not 1-to-1 functions. Anyway, let's use an example integral...
  31. FeDeX_LaTeX

    Solving a Differential Equation with a Substitution

    I have the differential equation: 4(2x^2 + xy) \frac{dy}{dx} = 3y^2 + 4xy The only thing I could see working is a substitution, but I can't work out which one to use. I've tried letting v = xy, or v = y/x, but neither of those seem to produce anything useful. Can anyone give me a hint?
  32. jaumzaum

    Substitution vs Elimination on halides

    If you add a strong base to a halide, you get mostly the alkene. If you add a weak base, especialy on primary halides not branched on the β carbon, the product is mostly the substituted. Why is that? 1) The mechanism for the substitution reaction is the heterotytic break of the C-X (where X is...
  33. Mathelogician

    MHB A question on substitution in predicate logic

    Hi everybody! I am confused about what is the role of the condition " xdoesn't belong to FV(phi)" in theorems like (i),(ii) or similarly in (iii) and (iv) . I know that the philosophy of the condition "the variable z's being free for x in phi" is to avoid the phenomenon that a free variable turn...
  34. F

    Trigonometric Substitution 11x^2dx/(25-x^2)^(3/2)

    Homework Statement Integral (11x^2)/(25-x^2)^(3/2) dx from 0 to (5*sqrt(3))/2 Homework Equations sin^2(θ) = 1 - cos^2(θ) The Attempt at a Solution 1. Factor out 11 from integral for simplicity. 11 * integral (x^2)/(25-x^2)^(3/2) 2. Re-write denominator of integral to...
  35. C

    Integrating Fractions with Substitution

    So the problem is ∫(6x+5)/(2x+1)dx. I know the proper way to solve this is to long divide these two expressions and then solve. However, I tried doing it with substitution. u = 2x+1 dx = du/2 I then reasoned that 3u + 2 = 6x+5 since 3(2x+1) + 2 = 6x+3+2 = 6x+5 so I substituted it on top...
  36. C

    Using this substitution, show that this integral is equal to the RHS

    Homework Statement Using the substitution x=acos^2(\theta)+bsin^2(\theta) \int^{b}_{a} \sqrt{(x-a)(b-x)}dx = \frac{\pi}{8}(b-a)^2The Attempt at a Solution After making the substitutions and doing all the algebra, I have \int^{\frac{\pi}{2}}_{0} sin(\theta)cos(\theta)(b-a)dx, with...
  37. K

    Integration By Substitution Problem (Trig)

    Homework Statement Integrate the following using substitution techniques ∫e3tcsc(e3t)cot(e3t) dt Homework Equations csc(t) = 1/sin(t) cot(t) = 1/tan(t) cot(t) = cos(t)/sin(t) 1 + cot2(t) = csc2(t) The Attempt at a Solution ∫e3tcsc(e3t)cot(e3t) dt set u = cot(e3t)...
  38. S

    Nucleophilicity in substitution reactions

    Is there any absolute order of the nucleophilicity of nucleophiles participating in organic substitution reactions or is it dependent on solvent,substrate or any other factors ? If so,how?
  39. C

    Integration using substitution

    Homework Statement I have a wire in the shape of a truncated cone. One side has radius a and the other has radius b. The wire has resistivity ρ and length L. I am supposed to find the resistance of the wire using R = ρL/A Homework Equations R= ρL/A The Attempt at a Solution So far I have...
  40. V

    Integration using sinh^2 substitution

    Homework Statement \int \frac{1}{\sqrt{x + x^2}} dx We have been told to use the substitution x = \sinh^2{t}.Homework Equations \int \frac{1}{\sqrt{a^2 + x^2}}dx = \sinh^{-1}(\frac{x}{a}) + C Maybe?The Attempt at a Solution I'm not really sure where to start, we haven't done any questions...
  41. M

    Finding the distance from a point to a plane using substitution

    If I want to find the distance from a point to a plane. E.g. (2,1,-1) to the plane x+y+z=1 I know that distance from one point to another is given by: \sqrt{(x-2)^{2}+(y-1)^{2}+(z+1)^{2}} And in this case the solution is to substitute in z=x+y-1 which fits nicely giving us...
  42. N

    How Can I Solve This Integral Using u-Substitution?

    ∫1/(10p-p^2)dp i tried using the u of substitution but for some reason I am unable to isolate dp and get an equation in terms of du which i could then plug into the integral and take the antiderivative.
  43. M

    How do I make this integral substitution

    If I want to find \int f(x) dx where x=a+b and a and b are both variables how would I do this in terms of a and b. Let me give you some context about this question. I'm trying to understand the following in my statistical mechanics class. My book states that if I have a random variable x...
  44. C

    Help with a simple cos substitution

    Homework Statement Hi folks, I am sure this is very simple but there are not enough steps given in this calculation for my simple brain to get from the beginning to the end! σ = ∫ (dσ/dΩ) = ∫ r2sin2θ (no integral limits given) σ = 2∏r2 ∫ (1 - u2) du (integral from -1 to 1) σ = 8∏r2 / 3...
  45. Z

    Integration U substitution then square it I think.

    1. ∫ 1/(2√(x+3)+x) 2. Not sure if I'm beginging this correctly or not but I get stuck. 3. Let u= √x+3 then u2 = x+3 2udu=dx dx=2√[(x+3) Therefore: ∫1/u2-3 Not sure where to go from here?
  46. D

    Trigonometric substitution in double integral

    Homework Statement Let R = \{ (x,y) \in \mathbb{R^{2}}: 0<x<1, 0<y<1\} be the unit square on the xy-plane. Use the change of variables x = \frac{{\sin u}}{{\cos v}} and y = \frac{{\sin v}}{{\cos u}} to evaluate the integral \iint_R {\frac{1} {{1 - {{(xy)}^2}}}dxdy} Homework Equations...
  47. A

    Simple Region Question for a Double Integral Substitution

    Homework Statement Evaluate the double integral integral ∫∫2x^2-xy-y^2 dxdy for the region R in the first quadrant bounded by the lines y=-2x+4, y=-2x+7, y=x-2, and y=x+1 using the transformation x=1/3(u+v), y=1/3(-2u+v).Homework Equations The Attempt at a Solution I've obtained the Jacobian...
  48. W

    Area & Centroid of Region Bounded by arcsinx & x=1/2

    Homework Statement consider the region bounded by the graphs of y=arcsinx, y=0, x = 1/2. a) find the area of the region. b) find the centroid of the region.Homework Equations \displaystyle\int_0^{1/2} {arcsinx dx} u=arcsinx; du = \frac{1}{1-x^2}dxdv=dx ; v=x xarcsinx]^{1/2}_{0} -...
  49. H

    Integral with trig substitution

    Homework Statement ∫(x+1)/((x^2+1)^2) Homework Equations The Attempt at a Solution I have been able to separate this into 2 ∫x/(x^2+1)^2 dx which i found to be equal to (1/2)arctanx and ∫1/(x^2+1)^2 dx which i am unable to find What i did was sub in x=tanθ and dx=sec^2(θ)dθ, and with...
  50. W

    Evaluate Trig Subs Integral w/ e^x = sin∅

    Homework Statement Use a trigonometric substitution to evaluate the integral. Homework Equations \int e^x\,dx /\sqrt{1-e^2x} The Attempt at a Solution e^x = sin∅ x=lnsin∅ dx=cos∅/sin∅ \frac{sin∅*cos∅}{sin∅*\sqrt{1-(sin∅)^2}} \int sin∅cos∅ / sin∅(cos∅)\,d∅...
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