Substitution Definition and 816 Threads

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∅...

Homework Statement (2S,3S)-2-chloro-3-methylpentane + high concentration of CH3O- ---> ? The Attempt at a Solution I need to find the product and usually my method is to knock off the -Cl replacing it by -OCH3 and when drawing the compound I just make an mirror image. In...

I suppose you all know the substitution formula for integrals. Well sometimes it seems to me you use substitutions which just don't fit directly into that formula. For instance for the integral of 1/(1+x^2) you substitute x=tan(u). Why is it suddenly allowed to assume that x can be...

Homework Statement evaluate the integral. Homework Equations integral (x^3 / (x^2 - 16) The Attempt at a Solution x=4sec∅ dx=4sec∅tan∅d∅ 1. i substituted those values in, and then split sec^4∅ into sec^2∅ and (1+tan^2∅). 2. integral 16 (1/u) du + integral 16 (u) du. 3. end...

Homework Statement ∫xtan^-1(x^2)dx Homework Equations The Attempt at a Solution I did u = x, du = 1, v = ? ,dv = tan^-1(x^2)dx I do not know how to get the integral of tan^-1(x^2)

Homework Statement Making sure I got the right answer.Homework Equations The Attempt at a Solution

Homework Statement integral (1)/(x^2sqrt(36-x^2) Homework Equations The Attempt at a Solution I found X=6sinθ dx=6cos √(36-x^2)=√(36-sin^2θ)=6cosθ i think the problem is that i am not getting integral of ∫csc^2θ

Homework Statement integral of dx/((9-(x^2))^(3/2)) A = 0, B = 3/2 Homework Equations Trigonometry Substitutions 3. The Attempt at a Solution : I am stuck with this question. So far, I got (1/9)integral of (1/cos^2(θ)) dθ

If I had an integral \int_{-1}^{1}e^{x}dx Then performing the substitution x=\frac{1}{t} would give me \int_{-1}^{1}-e^\frac{1}{t}t^{-2}dt Which can't be right because the number in the integral is always negative. Is this substitution not correct? Sorry if I am being very thick but I...

Homework Statement I am given an integral for which I need to substitute variables to remove a singularity so that the integral can be computed in Matlab using the Composite Trapezoidal Method, and then compared to the integral computed in Maple to 16 digit precision. I am stuck on the variable...

What is the easiest way to take the integral of: \int\frac{(6+e^{x})^{2}dx}{e^{x}} I have been having quite some difficulties with this one but here is my work so far: Let u=e^{x}, du=e^{x}dx =\int\frac{(u+6)^{2}du}{u^{2}} Then let s=u+6 ∴ u=s-6, ds=du =\int\frac{s^{2}ds}{(s-6)^{2}}...

The problem statement Evaluate the indefinite integral ∫\frac{\sqrt{x}}{\sqrt{x}-3}dx The attempt at a solution My first thought was to substitute u for √(x)-3, but then du would equal \frac{1}{2\sqrt{x}}dx, and there's no multiple of du in the integrand. Next, I tried splitting up...

I am unsure whether I have properly performed the integration of the integral ∫((sin(√x))^3*dx)/√x When I used my TI-Nsprire CAS to take the derivative of my answer in order to check if I was correct, and it came out differently. Now I used some trig identities to manipulate the problem, so I...

Homework Statement evaluate the definite integral ∫(0 to 3) dx/sqrt(25+x^2) Homework Equations The Attempt at a Solution I first used substitution and set x=5tanθ, and dx=5tanθsecθdθ then i wrote the integral as 5∫ tanθsecθdθ/sqrt(25(1+tan^2(θ)) after some simplification i...

Homework Statement \int \frac{dx}{\sqrt{x^{2}+16}}Homework Equations The Attempt at a Solution x=4tan\theta dx=4sec^{2}\theta d\theta Therefore: \int \frac{4sec^{2}\theta d\theta}{\sqrt{16tan^{2}\theta +16}} = \int \frac{sec^{2}\theta d\theta}{\sqrt{tan^{2}\theta+1}} \int \frac{sec^{2}\theta...

Homework Statement Find the following integral ∫1/(x*sqrt(x^2-1) dx Homework Equations The Attempt at a Solution I've decided to use the substitution: x = sec u dx = sec u * tan u du Substituting on the integral I got: ∫sec(u)*tan(u) / [sec u * sqrt((sec^2(u)-1))]...

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...

Homework Statement \int_{-p}^{p} \frac{2p}{(1+v^2)\sqrt{p^2 + v^2 +1 }} dv Homework Equations 1 + \tan{\theta}^2 = \sec{\theta}^2 The Attempt at a Solution I thought the best way to go about this was to rename some constants. Let \alpha^2 = 1 + p^2 so that we are left with...

If we see the form \sqrt { { a }^{ 2 }-{ x }^{ 2 } }, we always set x=asinθ How do we know that it will work in advance? Just trial & error?

It's been a year since I took Calc I, and I'm taking Calc II online this semester. This is technically a review problem from Calc I, and I managed the other seven, but I can't figure out how to solve this problem. 1.a Homework Statement ∫(a*sin(14x))/(\sqrt{1-196x^2} dx, evaluated at x=0...

Homework Statement I do not know how to solve the following indefinite integral. I personally think it is very difficult and would appreciate it had someone can explain it step by step? Homework Equations / The Attempt at a Solution This integral must been solved by mix of...

Homework Statement http://i.imgur.com/d0EKw.png Initigral (x* dx/((1+x^2)^.5) substitution u = 1+x^2 du = 2xdx how do get this to equal the inigral of U^-1/2 I am drawing a blank for the numerator I know how do the problem after I get u^-1/2 .. but i need to know how to get...

Studying for finals here...So I have this specific problem to use trig substitution on. $$\int \frac{x^2}{\sqrt{1-x^2}}\,dx$$ I begin by substituting $$x={sin{\theta}}$$ I am fine with doing everything up to the point where I have an answer for the integral in terms of \(\theta\). This...

Hi, I want to solve an overdetermined non-linear equation with the method of least squares. Assume it's f(x) = 1 + ax + a^2 + b, and I want to estimate a and b. This is non-linear, as I said, so the derivatives of the squared residuals involve a^3 terms and are difficult to solve. Now I thought...

Homework Statement 5a = 5 - b 5a = 3 - b Homework Equations The Attempt at a Solution I got the solution set to be 1/2, 5/2 i used substitution for substituting a into b of the second equation, just like they were x's and y's just used a's and b's there is no difference...

Homework Statement Integrate -1/(1+x(sin(t))^2) between 0 and pi/2 using the substitution u = tan(t)The Attempt at a Solution du/dt = (sec(t))^2 dt/du = 1/(1+u^2) I've messed around with the integral and trig. identities but I don't seem to be getting anywhere changing the integral to make...

Homework Statement I have four equations and have four variables. I need to solve for each of the variables. I am having difficulty figuring out how to do this. My equations are here. http://imgur.com/EOA8I Homework Equations \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 The Attempt...

Hi guys... I'm probably missing something pretty basic here but I can't seem to figure this out. I was working on a problem recently: for the complex functions f(z)=ez and g(z)=z, find their intersections. This post is not about the problem, it is about something I noticed while tackling it...

I have a problem with Double Integral that I can not seem to get correct. 4 2 ∫ ∫ e^(y^2)dydx 0 (x/2) The answer is (e^4)-1, but I can't seem to get the Substitution at all right. I have literally spent hours on this problem. Any help would be greatly appreciated, its...

Homework Statement I'm stuck at an attempt to solve an integration step. I think I'm supposed to trig substitute? Homework Equations http://img685.imageshack.us/img685/9158/unavngivetn.png It is the second to third equation I'm having a hard time with The Attempt at a Solution From second...

integration by parts I'm working through Apostol's Calculus. I have attached the problem. I need to derive the formula integrating by parts. It is not a hard problem, but I can't seem to understand how on Earth the author came up with that expression. I take f(x) = (a^2 - x^2)^n, so...

Homework Statement Show that y(t) = (1/w) ∫[0,t] f(s)*sin(w(t-s)) ds is a particular solution to y'' +w2 y = f(t)where w is a constant. The Attempt at a Solution After wasting several pages of paper I have made virtually no progress. Obviously, substitution suggests you plug in y(t)...

μ^{2}\frac{d^{2}u}{dx^{2}}+ae^{u}=0 Boundary conditions: u(-L)=u(L)=u_{0} Solve by multiplying by \frac{du}{dx} and integrating in x I know you have to use substitution, but I keep going in circles.

Homework Statement From Larson, 9th Edition: Section 4.5. Solve the differential equation \frac{\operatorname{d}y}{\operatorname{d}x}=4x+ \frac{4x}{\sqrt{16-x^2}}Homework Equations The Attempt at a Solution Well, I can get my book's answer, but not through doing things in the prescribed way...

Homework Statement use the substitution u= x+y and v=y-2x to evaluate double integral from ∫1-0∫(1−x) -(0) of (√x+y) (y−2x)^2 dydx Homework Equations integration tables I am assuming The Attempt at a Solution i tried to integrate directly but none of my integration tables match...

Hi! I am currently working with a linear PDE on the form \frac{\partial f}{\partial t} + A(v^2 - v_r^2)\frac{\partial f}{\partial \phi} + B\cos(\phi)\frac{\partial f}{\partial v} = 0. A and B are constants. I wish to find a clever coordinate substitution that simplifies, or maybe even...

We have a gravitational force on Planet X F=mγy^2 and we want to know the particle's final velocity. I know how to get the right answer, but I am wondering how come this doesn't work. F = mγy^2 ma = mγy^2 ∫a dt= γ∫y^2 dt Integral from 0 to t, I take v_0 = 0 y_0=0 v = γy^3/3

Homework Statement By using substitution u=\frac{1}{t}, or otherwise, show that \int^∞_1 \frac{t^5}{(1+t^3)^3}dt=\int^1_0 \frac{u^2}{(1+u^3)^3}du Homework Equations The Attempt at a Solution Well, the reverse can also be done (making t to u). However, I don't know how to...

the question is ∫dx/ x^2*√(x^2-1) I use x=a sec ∅ x^2*√(x^2-1)= sec^2∅tan∅ x=sec ∅ dx= sec∅tan∅d∅ so it will become something like this ∫sec∅tan∅d∅/sec^2∅tan∅= ∫1/sec∅d∅=∫cos∅d∅ =sin∅+c But how can i change this sin in...

∫(cot^4 x) (csc^4 x) dx Wolfram wants to use the reduction formula, but I'm meant to do this just using identities and u substitution. I was thinking something along the lines of: =∫cot^4 x (cot^2 x + 1)^2 dx =∫cot^8 x + 2cot^6 x + cot^4 x dx but I don't know where to go from there.

Homework Statement You know the U substitution proofs for inverse trig functions that go like this: \int\frac{1}{a^{2}+x^{2}}dx \int\frac{a\frac{1}{a}}{a(1+\frac{x^2}{a^2})}dx let u = x/a du= dx/a ... \frac{1}{a}tan^{-1}(x/a)+cI have searched google and can't find any of these proofs for...

Homework Statement ∫3xdx/√(1-2x) Homework Equations The Attempt at a Solution so i tried making u=3x which makes du=3dx but that substitution doesn't get rid of the x unde the square root. i tried u=1-2x and that gives du=-2dx and that doesn't get rid of the x on top. So I'm...

Find ∫from .6 to 0 x^2/ sqrt(9-25x^2) dx My teacher worked this on the board a little confused O obviously the trig sub is asintheta. But it isn't in the right form yet. So get it there you pull out a 25 --) sqrt(25(9/25) - x^2 ) 5sqrt((9/25)-x^2 so x= (3/5) sintheta so dx = 3/5costheta. so...

The integral from 0 to pi/2 of: cos(t)/sqrt(1+sin^2(t)) dt I'm supposed to use trig. substitution to find the solution. I started by using the formula a^2+x^2 to get x=atanx. In this case, sin(t)=(1)tan(θ), and so cos(t)dt=sec^2(θ)dθ and so I substituted this into the equation and got...

Hi, I have the equation y' = y^2 + x^2 and am asked to linearise the equation with the appropriate substitution and then solve the resulting 2nd order linear equation. My issue is I am unsure what to substitute in for y. I can't seem to find a choice for y which the differential will be a...

Homework Statement \int\frac{1}{\sqrt{16-x^2}}dx Homework Equations csc\theta=\frac{4}{\sqrt{16-x^2}} 4cos\theta=x -4sin\theta d\theta=dx \theta=arccos(\frac{x}{4}) The Attempt at a Solution Using these facts, I concluded that the integral, after all of the substitution, was...

Homework Statement Given that n is a positive integer, prove ∫sin^n(x)dx=∫cos^n(x)dx from 0 -> pi/2 Homework Equations Perhaps sin^2(x)+cos^2(x)=1? Not sure. The Attempt at a Solution I honestly don't even know where to start. Should I set u=sin(x) or cos(x)? Doesn't seem to get...

Homework Statement ∫(4x^3)/√(x^2+4)dx Homework Equations The Attempt at a Solution So, I let x= 2tanθ dx= 2sec^2θ dθ So, √(4tan^2(θ)+4)=2secθ ∫(4x^3)/√(x^2+4)dx=∫((32tan^3(θ))/(2secθ))2sec^2(θ)dθ. Would it go to ∫16tan^3(θ)2sec(θ)dθ or ∫32tan^3(θ)sec(θ)dθ

When you have to integrate a function that requires substitution and you integrate it again, why is it wrong to keep the initial substitution? e.g. y''=2x/(1+x^2)^2 If you let u=1+x^2 then y'=-(1/u)+C. Why is it wrong to integrate that again with respect to u and then change back to x at...

Homework Statement Evaluate the following indefinite integral: ∫(sin(ln16x))/xdx Homework Equations The Attempt at a Solution let u = ln16x therefore du=16/16x=1/x ∫sinudu =-cosu =-cos(ln16x) Why is this showing as the wrong answer?