# What is Substitution method: Definition and 23 Discussions

In optical fiber technology, the substitution method is a method of measuring the transmission loss of a fiber. It consists of:

using a stable optical source, at the wavelength of interest, to drive a mode scrambler, the output of which overfills (drives) a 1 to 2 meter long reference fiber having physical and optical characteristics matching those of the fiber under test,
measuring the power level at the output of the reference fiber,
repeating the procedure, substituting the fiber under test for the reference fiber, and
subtracting the power level obtained at the output of the fiber under test from the power level obtained at the output of the reference fiber, to get the transmission loss of the fiber under test.The substitution method has certain shortcomings with regard to its accuracy, but its simplicity makes it a popular field test method. It is conservative, in that if it were used to measure the individual losses of several long fibers, and the long fibers were concatenated, the total loss obtained (excluding splice losses) would be expected to be lower than the sum of the individual fiber losses.
Some modern optical power meters have the capability to set to zero the reference level measured at the output of the reference fiber, so that the transmission loss of the fiber under test may be read out directly.

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1. ### Simultaneous equations substitution method

I'm really stuck on this one, I was able to get the answer but not by the substitution method. So its the weight as A and B so I get A + B = 24 A(3) = B(5) so in my head I calculate a few pairs, 3 x 5 = 15 but 3 + 5 only = 8 so the next pair would be 10 and 6 which is still to small so I move...
2. ### MHB Substitution Method to solve linear simultaneous equation

What I have done: I changed all fractions to common denom and that gave me 5y-5x=1 (1) *I numbered the fractions 5y+2x=5 (2) Then: 5y=5-2x Substitute into equation 1 (5-2x)-5x=1 5-7x=1 x=4/7 Thing is my answer says I should be getting x=0 Any hints?
3. ### Integration by substitution question

Homework Statement Question: To solve the integral ##\int \frac{1}{\sqrt{x^2-4}} \,dx## on an interval ##I=(2,+\infty)##, can we use the substitution ##x=\operatorname {arcsint}##? Explain Homework Equations 3. The Attempt at a Solution [/B] This is my reasoning, the function ##\operatorname...
4. ### Substitution method for finding an integral's interval changes

Look, I was wondering if substituting the variable more than once is valid and hence the definite integral intervals change this way. Consider the following integral (I'm working for finding the volume of a solid of revolution): *\pi \int_{-3}^{5}3^{2}-(\sqrt{\frac{y+3}{2}}+1)^2dy Personally I...
5. ### Complicated Integral Using the Substitution Method

Homework Statement Evaluate the following integral using a change of variables: \int\frac{dx}{\sqrt{1-\sin^4{x}}} Homework Equations If f(x)=g(u(x))u'(x) and \int g(x)dx = G(x) +C then \int f(x)dx = G(u(x))+C The Attempt at a Solution It seems helpful to first simplify a little to obtain...
6. ### Solve Wave Equation: e^(-x^2), x*e^(-x^2), -infinity<x<infinity

Homework Statement So it says solve this wave equation : [y][/tt] - 4 [y][/xx] = 0 on the domain -infinity<x<infinity with initial conditions y(x,0) = e^(-x^2), yt(x,0) = x*(e^(-x^2)) Homework Equations I used the D Alembert's solution which is 1/2(f(x+ct)+f(x-ct)) + 1/2c ∫ g(z) dz The...
7. ### Question about substitution method in integration

It is common that we replace \int u(x)v'(x)dx by \int udv where both u and v are continuous functions of x. My question is, must we ensure that u can be written as a function of v before applying this? The above substitution method is involved in the proof of integration by parts but I cannot...
8. ### MHB Help with an integral (substitution method)

Hello! I need help to integrate the square root of: p x2 + x4 by the method of substitution, p is just a constant. I've been trying for a long time but I can't get it right. I know the answer and it still doesn't help. Thanks! (sorry, I don't know the Latex stuff)
9. ### 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...
10. ### 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!
11. ### Stuck on one of the substitution method steps

I put it in std form, did the homogeneous test. it passed with degree 2. I substituted y=ux and dy=udx+xdu and now I'm stuck. it needs to be simplified somehow but I don't know if ux is one var or if it's u*x. Same goes for udx and xdu. Is it really u*dx+x*du? even assuming that is correct, it...
12. ### Substitution Method on indefinite integral

Say we are solving an indefinite integral ∫x√(2x+1) dx. According to the textbook, the solution goes like this. Let u = 2x+1. Then x = (u-1)/2. Since √(2x+1) dx = (1/2)√u du, x√(2x+1) dx = [(u-1)/2] * (1/2)√u du. ∫x√(2x+1) dx = ∫[(u-1)/2] * (1/2)√u du. <= What justifies this?? The...
13. ### HELP First Order DE using Substitution Method

I feel as if I have made the correct substitution, what am I missing? See Attachment. Thanks, Dane
14. ### Substitution method with Integration by Parts?

Substitution method with Integration by Parts? Homework Statement Evaluate the integral... ∫x^3[e^(-x^2)]dx Homework Equations ∫udv=uv-∫vdu The Attempt at a Solution I first tried using integration by parts setting u and dv equal to anything and everything. This seemed to make...
15. ### Solving x^2 + 2y = 9 w/ Sub Method

use the subtitution method to solve. x^2 + 2y = 9 x - y + 3 = 0 y=x+3 -> (3) substitute for y with x+3, you get: x^2+2(x+3)=9 x2+2x−3=0 factorize it to get: (x+3)(x−1)=0 x=−3,x=1 substitute for each value of x in equation 3 x=−3y=−3+3=0 x=1y=1+3=4...
16. ### Differential equations substitution method

Homework Statement y'=y+y^3 Homework Equations The Attempt at a Solution I set y=v, dy/dx = dv/dx. Substituted back into original equation ST dv/dx = v + v^3. Cross multiply, then divide yielded dv/(v+v^3) = dx. After that, I have no clue. The book gives the following...
17. ### Solving an integral by substitution method

Homework Statement Hi there. I'm dealing with undefined integrals now. And I found this one that I don't know how to solve. The problem statement says: Solve the next integrals using the substitution method. \displaystyle\int_{}^{}\displaystyle\frac{\cos(x)}{\sin^3(x)}The Attempt at a Solution...
18. ### How to Integrate [1/(x^2 + 1)] dx?

Hi everyone, Can you tell me how to integrate the following equation? \int\frac{1}{x^2 + 1} \ dx I've tried the substitution method, u = x^2 + 1, du/dx = 2x. But the x variable is still exist. Also, the trigonometry substitution method, but the denominator is not in \sqrt{x^2 + 1}...
19. ### 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(...
20. ### Ode substitution method

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...
21. ### 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...
22. ### The Substitution Method

Evaluate the following integral using integration by substitution: http://img254.imageshack.us/img254/750/44900023cm4.png [Broken][/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 =...
23. ### Solve S (x+5)½/x-4 dx with Substitution Method

Hi, I have another problem about substitution Method. I think this method is used to make the problem to solve in easy way but it is making my procedure too long for this problem. Can you solve it by substitution method. S (x+5)½/x-4 dx where S is the sign of integral. The answer of...