Substitution Definition and 797 Threads

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)...

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...

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...

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...

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.

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...

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}}...

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...

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 =...

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...

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...

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) =...

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)...

Homework Statement see attached Homework Equations The Attempt at a Solution

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 =...

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...

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...

\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...

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}~~ -...

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}...

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.

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

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 !

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...

\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...

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...

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...

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 +...

[SIZE="4"]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)...

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...

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

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...

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

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...

\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(...

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...

Homework Statement I am trying to solve a line integral (bear with me, I am new to calculus!) and my basic skills of integration seem to fail me. I am sure the mistake is quite obvious, as I keep getting the wrong answer, 2, when it should be ~2.69 Homework Equations \int_C 4x^3dS C is the...

ok i have been studying the in-depth processes of trigonometric substitution with integrals and this problem has me frusterated. \int x^2\sqrt{(x^2-4)} dx The evaluation is clear (from an old Table of Integrals I found), but the derivation is not at all clear, which is what i want to know. I...

Homework Statement Let's say you make the inverse substitution x = 2tan(z) in some integral. Let's say you evaluate the integral get something like like 4sin(z). How do you put z back in terms of x? Homework Equations The Attempt at a Solution I can do it by saying arctan(x/2) =...

Homework Statement Evaluate: \int\\{1}/{\sqrt{x^2-1}} dx between -3, -2 I know I'm supposed to use hyperbolic substitution in the question. Homework Equations edit: cosh^2(t) - sinh^2(t) = 1 The Attempt at a Solution let x = -cosht, inside the integral let dx = sinh(t) dt int (...

I’ve got this tricky trig substitution that I’ve been trying to do as for my Calc II class. I think that I’ve done the substitution part right I just have no idea where to go from here. I’ve tried many routes including integrating by parts as well. Please help me thank you...

Homework Statement Use substitution to evaluate the integral. \int \frac{4cos(t)}{(2+sin(t))^2}dt Homework Equations None, really.The Attempt at a Solution I'm not sure what to use as u, for the substitution. I've tried (2+sin(t))^2, as well as other attempts, but I can't seem to find anything.

Homework Statement Integrate: \int\sqrt{1 - 9t^{2}}dt Homework Equations The Attempt at a Solution t = 1/3 sin x dt/dx = 1/3 cos x dt = 1/3 cos x dx 3t = sin x 1/3 \int\sqrt{1 - sin^{2} x} cos x dx 1/3 \int cos ^{2}x dx 1/3 \int(1 + cos 2x) / 2 dx 1/6 \int1 + cos 2x dx...

Homework Statement Integrate: \int\sqrt{1-9t^{2}}dt Homework Equations The Attempt at a Solution t = 1/3 sin\Theta dt/d\Theta = 1/3 cos\Theta dt = 1/3 cos\Thetad\Theta 3t = sin\Theta 1/3\int\sqrt{1-sin^{2}}\Theta cos\Thetad\Theta 1/3\int cos^{2}\Thetad\Theta 1/3\int(1 +...

I tried to use the latex reference but it was way too complicated for me. Can you see the attachment. Thank you!

Integration by parts and substitution help! Homework Statement ∫0 -1 e ^√x+1 Homework Equations The Attempt at a Solution

Homework Statement Question: How many simple substitution ciphers are there where no point is fixed (ie: no letter is mapped to itself)? EDIT: Incase the termonoligy is different, a simple substitution cipher is a mapping where the plaintext in english is encoded so that every letter is...

Homework Statement Solve the following differential equation by using substitution. Homework Equations (x + 2y - 1)dx + (2x + 4y - 3)dy = 0 Let u = x + 2y The Attempt at a Solution u = x + 2y du/dx = 1 + 2(dy/dx) (x + 2y - 1)dx + (2x + 4y - 3)dy = 0 (u - 1)dx + (2u -...

Homework Statement dy/dx = (x+3y)/(3x+y) I have to solve the given differential equation by using an appropriate substitution... The Attempt at a Solution I used algebra to make the equation (3x+y)dy - (x+3y)dx = 0 then made x = vy and dx = vdy + ydv. then plugged into get...

Homework Statement Lim [(tanx)^2] / [1 + secx] <<< as x goes to pi Homework Equations The Attempt at a Solution (tan x)^2 = (sin x)^2 / (cos x)^2 (sin x)^2 = y lim y = 0 <<< as x goes to pi lim [y/ (cos x)^2] / [1 + (1/y)] <<< as y...