Substitution Definition and 816 Threads

Homework Statement I've done 1-3 and 10-20, but these give me an extreme headache. Assume I know everything there is to know about integration and the trigonometric and hyperbolic functions. Change the following into variables with x with... 4. (1/4) * sin(z) - (1/12) * sin^3(z) after...

$\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

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

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

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??

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

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...

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 $81\int\sin^2{\theta}\cos{\theta}\sin{\theta}\ dx$...

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

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...

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}}...

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

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...

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...

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...

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...

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}...

∫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...

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.

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...

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...

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...

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...

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 =...

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 ?

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)...

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...

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!

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.

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...

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...

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?

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...

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...

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...

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...

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...

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)...

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?

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...

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...

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...

∫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.

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...

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...

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?

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...

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...

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} -...

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...