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. A

    Integration by substitution with radicals

    Homework Statement 2 h(x)=∫√(1+t^3) dt find h'(2) x^2 Homework Equations The Attempt at a Solution I started out solving this equation by flipping x^2 and 2 and making the integral negative. From here on out, I'm lost. I've tried substituting u in for 1+t^3 and solving...
  2. H

    Solve Substitution Problem: Homework Equation v2=2v1

    Homework Statement d1/d2 + d2/v2 = 0.500km/v1 + 3.50km/2v1 = 0.250h Homework Equations v2 = 2v1 The Attempt at a Solution The answer is 9.00km/h but I keep coming up with 5.33. I know this is simple algebra but I am missing something in the order of operations. 0.500km/v1 +...
  3. M

    Substitution technique for integral

    Homework Statement I need to use a substitution technique and then the table of integration to integrate the following: ∫ x5 √(x4 – 4) dx and i am given a hint which is x5 = (x3)(x2) I would assume that u = x4 – 4, then du = 4x3 and that at some point a2 = 4 and a = 2 However...
  4. F

    Integration by Substitution: Simplifying Complex Integrals

    Homework Statement Compute the indefinite integral. ∫(x^2 + 1)^(-5/2) dx The Attempt at a Solution I have a hunch that I need to substitute x = tan(u) but, as always, my lack of trig skills are holding me back.
  5. E

    Easily Solve Substitution Problems with This Simple Guide

    http://i45.tinypic.com/2ue4g9j.jpg
  6. B

    Indefinite Integration with Logarithms and Substitution

    Hi, I missed a few days of my calculus class. I've managed to figure out how to use substitution to solve an indefinite integral, and can apply the log properties to some extent. I just can't figure out this problem. Homework Statement Find the indefinite integral: \int{\frac{1}{x...
  7. mg0stisha

    Making Integration Easier: Substitution for Tricky Integrals?

    Homework Statement \int sin^{5}x cosx dx Homework Equations None The Attempt at a Solution I tried setting u=sin^5(x) but this ended up yielding \frac{1}{5}\int u cos^{3}x du and I cannot think of a better substitution. Any tips?
  8. C

    Help figuring out this trigometric substitution

    From my notes I have w=u(x+iy)*(x^2 - y^2 +k^2 + i(2xy))^-.5 We let N=x^2-y^2+k^2 M=2xy R^2=(N^2+M^2)^2 theta=tan^-1(M/N) using this, now w=u(x+iy)*(cos(theta/2)-isin(theta/2))*(x^2 - y^2 +k^2 )^2 + (2xy)^2 )^-.25 I don't get that part. Btw, it simplifies to...
  9. S

    Solving Int. with Trig Substitution for Beginners

    Homework Statement \int\frac{\sqrt{1+x}+\sqrt{1-x}} { \sqrt{1+x}-\sqrt{1-x}}{dx} Homework Equations I believe trig substitution can be used here. I'm not very good at calculus only beginning to take calc classes, and guideance would be wonderful. because i want to get better. The...
  10. I

    Irreducibility of a polynomial by eisenstein and substitution

    Homework Statement Show 16x^4 = 8x^3 - 16x^2 - 8x + 1 is irreducible. Homework Equations Eisenstein's criteria, if there is n s.t. n does not divide the leading coefficient, divides all the other coefficients, and n^2 does not divide the last coefficient then the polynomial is...
  11. M

    Multiple integral with substitution

    Homework Statement This is going to be confusing to read, as I don't know how to make this look right. The first integral is from 0 to L-2d, the second from x1+d to L-d, and the third from x2 to L. (F(x)=1) 1.) 0\intL-2d,x1+d\intL-d,x2+d\intL dx3dx2dx1 2.) 0\intL-2d,x1+d\intL-d...
  12. M

    ODE with Substitution: Solving for y' = (2x+3y-5)/(x+4y) using z = b/a method

    Homework Statement y' = \frac{2x+3y-5}{x+4y} Homework Equations The Attempt at a Solution First of all, I switched it to another coordinates, a and b, giving: b' = \frac{2a+3b}{a+4b} where a = x-4 and b = y+1. Then using the substitution z = \frac{b}{a} and some algebra I...
  13. J

    Integration-Problem with substitution

    Homework Statement I have reached this integration from a mechanics problem about small angle scattering. t= (2pa/mv^2)*(int from p to infinity) [r*dr]/[((b^2 +r^2)^(3/2))(sqrt(r^2 - p^2))] Homework Equations The Attempt at a Solution I know that there should be a substitution...
  14. 3

    Integration by Substitution using Partial Fractions Decomposition

    Homework Statement Integrate \int\frac{dz}{1+e^z} by substitution Homework Equations The Attempt at a Solution I chose u=(1+e^{z}) so du/dz=e^{z} and dz=du/e^{z}. Therefore, \int\frac{1}{u} \frac{du}{e^{z}} I plug z=ln(u-1) in for z, so \int\frac{1}{u} \frac{du}{u-1}...
  15. T

    Integration with inverse trigonometric substitution

    Homework Statement The problem is: integrate: (1/(4x2+4x+5)2)dx Homework Equations The Attempt at a Solution 4x2+4x+5=(2x+1)2+4 gives: ∫dx/((2x+1)2+4)2 use regular substitution: u=2x+1 du=2dx dx=1/2du gives: 1/2∫du/(u2+4)2) trigonometric substitution: u=tan(z)...
  16. M

    Solve DE with Substitution: y' = cos(x-y)

    Homework Statement Solve: y' = cos(x-y) Homework Equations The Attempt at a Solution Using x-y=t and solving the integrals, I get that the general solution is: -cot(0.5(x-y)) = x + c which is correct, but there's another solution which is x-y=2πk, but I don't understand why...
  17. C

    Do I integrate it using Trigonometry substitution?

    Dear all, I would like to evaluate \int\frac{1}{1-2sin\left(x\right)}dx Firstly, I make use of the Weierstrass substitution method by letting: u=tan\left(\frac{x}{2}\right) and therefore sin\left(x\right)=\frac{2u}{1+u^{2}} and dx=\frac{2}{1+u^{2}}du Eventually I can...
  18. S

    Trig Substitution for Integrating (y-x)/(a^2(y)-a^2(x))^b - Homework Help

    Homework Statement I want to integrate: \int^y_0\frac{y-x}{(a^2(y)-a^2(x))^b}dx Homework Equations a2(y) means that a is a function of y. similarly for [a(x)]2. so [a(x)]2 is a functions that depends on x. The Attempt at a Solution I tried integration by parts: let u=y-x so...
  19. K

    Calculus 2: Trigonometric Substitution, using Z = tan(x/2)

    Homework Statement Homework Equations The Attempt at a Solution After substituting: Using I'm stuck here: I can't seem to find anything online about this substitution. Any help would be appreciated. thanks.
  20. K

    Substitution Differential Equation

    Homework Statement Solve the given initial value problem. Homework Equations ydx + x(lnx - lny - 1)dy = 0 The Attempt at a Solution I am confused as to what to do as I can't just substitute y=ux or x=vy because they are not homogeneous of the same degree. And I can't use exact because...
  21. K

    Solving a Diff Eq with Substitution

    Homework Statement Solve the given differential equation by using appropriate substitution. 3(1+t^{2})\frac{dy}{dx} = 2ty(y^{3} - 1) Homework Equations y = u^{\frac{1}{1-n}} The Attempt at a Solution \frac{dy}{dt} = \frac{2ty^{4} - 2ty}{3 + 3t^{2}} \frac{dy}{dt} + \frac{2ty}{3 + 3t^{2}}...
  22. I

    Find general solution to Bessel's equation with substitution

    Homework Statement 81x^{2}y'' + 27xy' + (9x^{\frac{2}{3}}+8)y = 0 Hint: y = x1/3u x1/3 = z 2. The attempt at a solution Change of variables gives: \frac{d^{2}y}{dx^{2}} = x^{\frac{1}{3}}\frac{d^{2}u}{dx^{2}}+\frac{2}{3}x^{-\frac{2}{3}}\frac{du}{dx} - \frac{2}{9}x^{-\frac{5}{3}}u...
  23. K

    Differential Equation Substitution

    Homework Statement Solve the given differential equation by using an appropriate substitution. Homework Equations x\frac{dy}{dx} = y + \sqrt{x^{2} - y^{2}}, x > 0 The Attempt at a Solution x\frac{dy}{dx} = y + \sqrt{x^{2} - y^{2}} xdy = (y + \sqrt{x^{2} - y^{2}})dx y = ux u =...
  24. S

    Integration w/ trig. substitution (x^2 + 1) / (x^2 - 2x + 2)^2

    Another even numbered problem in my book, so no textbook answer. I checked it in WolframAlpha(WA), but the answer came out slightly different. Hopefully no typos in this writeup. Homework Statement \int \frac{x^2 + 1}{(x^2 - 2x + 2)^2}dxHomework Equations I factor the denominator: (x^2 - 2x...
  25. L

    Nucleophilic acyl substitution and basicity of groups

    I'm studying the reaction mechanisms for carboxylic acid and its derivatives and here it says whether a compound with a C=O bond undergoes nucleophilic addition (as in aldehydes and ketones) or nucleophilic acyl substitution depends on the relative basicities of the substituent group. For...
  26. H

    Integration involving Substitution

    Homework Statement Find the arc length of the function f(x) = x(sqrt(x/2-x) from 0 to x Integral forms involving a+bu --> integral [sqrt(a+bu)]/u^2du Homework Equations Arc Length = integral sqrt(1+(f'(x))^2)dx The Attempt at a Solution First, I took the derrivate f'(x) =...
  27. S

    Solving First Order Differential Equation using substitution

    Hi, Here is the equation: x+x'=5.1sin(600*t)*u(t) Our teacher gave us a hint that we should try using a substitution which is a system of sines, cosines, and looks something similar to 5.1sin(600*t)*u(t). I tried substituting: x(t)= A sin (w1*t)+B cos (w2*t)+ c cos(w3*t)*u(t)...
  28. D

    Changing Limits of Integration affects variable substitution?

    Homework Statement see attached Homework Equations The Attempt at a Solution
  29. J

    Integration by Trigonometric Substitution

    Homework Statement Evaluate: \int\frac{1}{(4 - \tan^2{x})^{3/2}}dx Homework Equations I must integrate the above equation using only trigonometric subtitutions of algebraic equations. The Attempt at a Solution Here is what I have so far: Let \tan{(x)} = 2\sin{(\theta)} x =...
  30. S

    Why Is x/5 the Right Substitution for Integrating 1/(25+x^2)?

    Homework Statement I was asked to find the formula for the antiderivative \int1/(25+x^{2}) Homework Equations Take a 'part' of the equation and use it to solve the antiderivative, integration by substitution. and dw=(1/5)dx The Attempt at a Solution I initially set my substitution...
  31. S

    Integrating Difficult Equations: U Substitution

    Homework Statement As I was reviewing some of my previuosly learned calculus I came across somthing that I had either forgoten how to do or was never taught. How do you take the integral of somthing like this \int \sqrt{\frac{9}{4}x+1} I don't know how to start with this one. Do I use U...
  32. P

    Variable substitution question(diff)

    \frac{xy'}{(\ln x\arctan y)-1}=(1+y^2)\arctan y\\ t=\arctan y\\ t'=\frac{1}{1+y^2}y'\\ \frac{x}{\ln (x)t-1}=\frac{t}{t'}\\ \frac{x}{\ln (x)t-1}=\frac{tdx}{dt}\\ xdt=(\ln (x)t-1)tdx\\ \frac{dt}{\ln (x)t-1}=\frac{dx}{x}\\ still can't beak it as one type of variable on each side so i...
  33. James889

    U substitution and partial integration

    Hey, I need to evaluate \int_{1}^{5}(6-2x)\sqrt{5-x}dx So. \tex{Let ~~u} = 6-2x~~ \tex{then}~~ du = 2 \frac{1}{2}~du = dx New limits: x = 5 \longrightarrow u = -4 x = 1 \longrightarrow u = 4 now, -\frac{1}{2}\int^{4}_{-4} u*\sqrt{5-x} and now for the partial integration. u \sqrt{5-x}~~ -...
  34. P

    Differential equation and substitution

    This is a step from my notes that I don't follow. I have sin\theta\frac{d}{d\theta}(sin\theta\frac{d\Theta}{d\theta}) and that, when substituting u=cos\theta and writing that \Theta(\theta)=P(u), \frac{d}{du}((1-u^{2})\frac{dP}{du}) is obtained. I can see \frac{d\Theta}{d\theta}=\frac{dP}{du}...
  35. K

    Twelve Sum and Digit Substitution Puzzle

    Substitute each of the capital letters in the figure given below by a different digit from 0 to 7 such that: A+B+C = A+D+F = C+E+H = F+G+H = 12. A B C D E F G H Note: The rotations and reflections of a valid arrangement are deemed as the same solution.
  36. S

    Integral Substitution: Evaluating Int f(x4) dx

    How do I evaluate the integral \int_a^b f(x4) dx = \int_y^z f(u) dx/du du, where u=x4 y=a4 z=b4
  37. Q

    What Are the Correct u and du Values for These Trig Substitution Problems?

    I really needed some help with these trig substitution problems. Please write the u and du values for each. I'd really really appreciate it ! 1) Integrate [e^x/(e^x +1)] dx 2) Integrate [1/(e^2x +1)] dx 3) Integrate [(1+x)/(1+x^2)] dx Thank you Again !
  38. S

    Integration by Substitution: What Went Wrong in Solving \int x(x^{2}+5)^{75} dx?

    Homework Statement I've come across the problem \ \int x(x^{2}+5)^{75} dx Homework Equations The Attempt at a Solution Once going through the whole problem I got \ \frac{(x^{2}+5)^{76}}{76}+c but the text I have said the answer was, \frac{1}{152}(x^{2}+5)^{76}+c. What did I do...
  39. A

    Substitution to make it seperable

    \acute{y}+xy^{3}+\frac{y}{x}=0 y(1)=2 using substitution u=y^{-2} e^{y}\acute{y}=e^{-x}-e^{y} y(0)=0 using substitution u=e^{y} i could not make these equations seperable and solve for the IVP. Anyone has any idea? Edit: these problems are not homework, but for self study for preparation to...
  40. M

    Integration by Trig Substitution

    I've attempted solutions at this in two different manners and found myself stuck both ways. I'll show you the way that seems to make progress. The other way involves not factoring the junk under the radical in the denominator. Homework Statement \int^{1}_{0} \frac{3x^2 -1}{\sqrt{x-x^3}} dx...
  41. G

    Trigonometric Substitution for Evaluating Integrals: Using t=atan(theta) Model

    Homework Statement Use an appropriate substitution and then a trigonometric substitution to evaluate the integral: \int_-^{ln(4)} \frac{e^{t}dt}{\sqrt{e^{2t}+9}} Homework Equations Seems to be close to the t=atan(\theta) model \int \frac{e^{t}}{\sqrt{a^{2}+x^{2}}}dt The Attempt at a...
  42. B

    Integration by substitution find the area under the graph

    Homework Statement I have the function below which i need to find the area under the graph. Homework Equations \int_{ - \frac {\pi}{4}}^{\frac {\pi}{3}}\frac {2\sec x}{2 + \tan x}dx The Attempt at a Solution I can simplify it to \int_{ - \frac {\pi}{4}}^{\frac {\pi}{3}}\frac {2}{2\cos x +...
  43. D

    Solve Trig Substitution Homework with P

    Homework Statement If sin54=P, express the following in terms of p, without using a calculator. 1] cos18 2] \frac{tan27+cot63}{1+tan207.cot117} Homework Equations The Attempt at a Solution 1] cos18=sin(90-18) =sin(72) sin72=sin(180-108) =sin(180-108)...
  44. S

    Differential Equations - Substitution

    Solve the Initial Value Problem: dy/dx = (3x + 2y)/(3x + 2y + 2) , y(-1) = -1 I need to use substitution and use the formulas: du/dx = A + B(dy/dx) dy/dx = (1/B)*(du/dx - A) Let u = 3x + 2y then du/dx = 3 + 2(dy/dx) therefore A = 3, B = 2 Pluging it into the formula gives me...
  45. P

    What is the Substitution Method for Integrating a Rational Function?

    Homework Statement ∫1/(x^2+2x+2) dx Homework Equations The Attempt at a Solution u = x^2+2x+2 du = 2dx(x+1) But I am left with an x and can not find the antiderviative
  46. Z

    How can I use integration by substitution to solve this equation?

    Homework Statement http://img20.imageshack.us/img20/112/41590752.jpg Homework Equations The Attempt at a Solution I have no idea how to convert the left equation into the right one. Could someone show me how to do that? I don't understand why the right equation should be...
  47. N

    Efficient U-Substitution for (x^2)(sinx)/(1+x^6)

    [ ( x^2 ) ( sinx ) ] / (1 + x^6)
  48. C

    Periodic function and substitution question

    Let f: R-> R be a continuous function. Let T>0 be such that f(x+T)= f(x) for all x. We say that f is a periodic function with a period T>0. Use an appropriate substitution to prove that for all real numbers a \int^{a+T}_{a}f(x)dx = \int^{T}_{0}f(x)dx. I have no idea how to do this...
  49. tony873004

    Differential equation substitution method

    \begin{array}{l} \frac{{dy}}{{dx}} = \frac{{4x^2 + 5xy + y^2 }}{{x^2 }} \\ \\ \frac{{dy}}{{dx}} = 4 + \frac{{5y}}{x} + \left( {\frac{y}{x}} \right)^2 \\ \\ {\rm{Let }}v = y/x\,\,\,\, \Rightarrow \,\,\,\,y = vx \\ \\ \frac{{dy}}{{dx}} = 4 + \frac{{5v}}{{xx}} + \left(...
  50. 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...
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