Definite integral Definition and 389 Threads

Homework Statement this is definite integral question. lower limit 0 ; upper limit 1 ; integral (e^x)(x-1)^n = 16-6e find n (n<6) here e is euler constant value around 2.7 (irrational) hope you understood. Homework Equations as much as first year student know. The Attempt at a Solution...

Homework Statement \int^{\frac{\pi}{4}}_{0}\frac{1+cos^{2}(x)}{cos^{2}(x)} Homework Equations The Attempt at a Solution I tried u-sub, splitting it up into 2 fractions, and a trig identity (cos^2x=(1+cos2x)/2). Am i missing something? The other problems were quite easy.

Is it true that \int_0^1 f(x) dx \in \mathbb{Q} \Rightarrow \int_0^1 x f(x) dx \in \mathbb{Q} ? (Suppose f(x) integrable as needed) I thought of this conjecture yesterday and still couldn't prove it, I tried using integration by parts to relate it to the original, but didn't...

Homework Statement Use 2 or more substitutions to find the following integrals hint : begin with u=cosx Homework Equations Integral 0--->pi/2 (cosxsinx)/swrt(cos2x+49 dx The Attempt at a Solution I'm still a little fuzzy on using multiple substitutions. From what I've read on...

I'm wrestling with this problem and I'd appreciate any pointers anyone can give me. (I've tried Google, but no luck.) I'm trying to figure out the total mass, with respect to the Haar measure I guess, that's accounted for by a definite subset of \mathbf{SO}(n). Specifically, working in...

1. Evaluate [PLAIN]http://www.goiit.com/equations/2011/5/25/795493e8-6a4d-42b0-8c18-2afa4e75b653.png I have tried hard to solve it but I am not getting the exact answer. The ans is given to be -a^2/(1+a^2)

i am struggling with this question for quite few days now and i am nowhere near to an appropriate solution. The previous ( related) question were regarding the the function g(x)= 5+9x-2x^2 /(x^2 -6x+23)^3/2. i have found the derivatives, the stationary point and also to classify them...

Homework Statement Hi guys, need help evaluating this integral. Integrate from -1 to 1 (t^i)(t^5-t^4) dt Where i just an arbitrary variable, (it's actually an index number of a matrix) Homework Equations The Attempt at a Solution So I thought I would begin by expanding...

Evaluate each Definite integral. ∫ lower limit 1, upper limit 8. 3 - y^(1/3) / y^(2/3) ; dy Here's a link if you have trouble understanding my problem. It's #24. http://pic20.picturetrail.com/VOL1370/5671323/23643016/396407299.jpg I did ∫(3*y^(-2/3) - y^(-1/3)) dy from 1 to 8 9*y^(1/3) -...

Use the definite integral to find the area between the x-axis and F(x) over the indicated interval. Check first to see if the graph crosses the x-axis in the given interval. F(x)= 4x-32 ; [5,10] I did y = 0 => 4x - 32 = 0 => x = 8 => graph crosses x-axis at x = 8 Also it is +ve for x > 8...

Hello, first time poster. I am putting this question in the PDE section because this was a question I came up while solving a PDE question. Also, I figured that since this is not a straight homework question, I can post in this category. Mods, feel free to move this post to wherever you...

Given that 1/x is symetric across y=x, why can't we say \int^1_0 1/x - x dx= \int^\infty_1 1/x + x dx? Geometrically, it makes sense, but ln(0) is clearly undefined.

Homework Statement п/2 [SIZE="6"]∫dx/(2+sinx)2 0 Homework Equations The Attempt at a Solution п/2 [SIZE="6"]∫dx/(2 + sinx)2 = 0 п/2 ∫dx/(4 + 4sinx + (sinx)2) 0 substitude t=tg x/2 => x=2arctgx dx=2/(1+t2) dt sinx=2t/(1+t2) t=tgx/2 => 1 [SIZE="6"]∫(1/(4 + 4*(2t/(1+t2))...

Homework Statement π/2 d x/ [SIZE="6"] ∫ (x2 - 2x cosx  + 1) 0 п [SIZE="6"]∫ sin x. ex^2dx -п Homework Equations The Attempt at a Solution

If we have a definite integral given by f(x) = ∫sintdt with limits from 0 to x, can we integrate the function directly by substituting t as x because the variable involved in the integrand is a dummy variable?

[PLAIN]http://img600.imageshack.us/img600/1210/11096142.png Hey I was wondering if you guys could help me out with this question... I think I have the right power series: = \frac{1}{1-x} + \frac{x}{1-x} = (1+x+x^{2}+x^{3}+...)(x+x^{2}+x^{3}+x^{4}+...) = 1+2x+2x^{2}+2x^{3}+2x^{4}+... = 1 +...

Can somebody give me an example of a definite integral satisfying this property? :

Homework Statement Evaluate. \int y2(1 - y3/a2)-2 dy Upper limit = 0 Lower limit = -a Homework Equations The Attempt at a Solution The a's make me really confused and I have no idea how to even start this question. What I started with was: Set u = 1 - y3/a2 du =...

Homework Statement Evaluate integral from -1 to 1 of: sqrt(ln(6-x))/(sqrt(ln(6-x))+sqrt(ln(6+x))) Homework Equations n/a The Attempt at a Solution I barely know how to approach this integral. I've been trying to figure this out for a long time now, and I feel like I haven't...

[PLAIN]http://img25.imageshack.us/img25/8933/lastscante.jpg I am new to integration by parts and am not sure what boundries to use when eveluating v on the bottom right.

Homework Statement The definite integral of from 0 to 1 of ∫ (r3)dr/sqrt(4+r2)Homework Equations ∫udv = uv - ∫vdu ∫du/sqrt(a2 - u2) = arcsin(u/a) + C ∫du/(asqrt(a2 - u2)) = (1/a)arcsec(u/a) + C The Attempt at a Solution I made u = (4+r2)-1/2 because I thought it easier to get it's...

Homework Statement I am getting the answer as -(pi) The answer given is (pi). There might be a sign error. I checked it many times but cannot find my fault. I need some help.

The integral I am seeking to evaluate is int_{-\inf}^{+\inf) exp(-|w|/2) exp( i w [ln(|w|)/pi-x]) dw a definite integral which is a function of x... at x=0, it is nearly the same as the "sophomore dream" integral... I don't know why, but this function seems to have a singularity at...

Homework Statement The problem: Find the derivative of x2 multiplied by the integral of dt/(t-2t2) from x to -x2+2x i.e, the function is x^2 times integral of [1/(t-2t^2)]dt from a to b with a being x and b being -x^2+2x Homework Equations Derivative and integral definitionsThe Attempt at...

Homework Statement A cylinder mug with a 4 centimeter radius and 8 centimeter height is filled with tea. When sugar is added to the tea, the sugar usually settles to the bottom of the cylinder mug. The density of sugar in the tea at a height h centimeter from the bottom of the cylinder mug is...

Hi everyone, to day I have a problem in dealing with the following question: and I hope that you help me here is may question and my try: [PLAIN]http://img268.imageshack.us/img268/9291/scan0004ov.jpg wish you all the best

Homework Statement Integrate from 0 to sqrt(pi/2). I can't figure out how to make the LaTeX work correctly for it, so it just looks like an indefinite integral. \int 7\theta^3cos(\theta^2) d\theta Homework Equations \int udv = uv - \int vdu The Attempt at a Solution First...

I was reading on a textbook and came across a step and I had no idea how the author got the next step from it. This was basically differentiation of a definite integral. So here it is. \frac{\partial}{\partial t}{\int_0^t}{\int_0^1}u_t^2(x,s)dx ds So I figured that u_t=\frac{\partial...

Homework Statement Q) Evaluate The Definite Integral: ∫ (x^3) / (1 + x^4) dx Upper Limit: 1 Lower Limit: 0 Homework Equations The Attempt at a Solution I think I'm on the right track; u = 1 + x^4 du/dx = 4x^3 du = 4x^3 dx 1/4 du = x^3 dx When x = 0 ; u =...

This is not really a homework. Homework Statement I am trying to solve one definite integral Homework Equations \int_0^{2 \pi} \frac{\sin^2{t}}{\sqrt{a\cos{t} + b}} dt where a, b are some positive numbers. The Attempt at a Solution I tried integrate by parts, also...

If my Force = (-gamma)(dx/dt) velocity v = dx/dt Mass m (m)(dv/dt) = F = (-gamma)(dx/dt) Now I want to integrate it over kT<= t <= (k+1)T Rearranging gives me (dv/v) = (-gamma/m)dt The right side integrates to (-gamma/m)(T), but how do I integrate the left side over kT<= t <=...

Homework Statement \int^{\infty}_{0} e^{-y} dy = 1 Homework Equations The Attempt at a Solution Why is this equality true? I understand that the integral and derivative of e^y is always e^y, but I can't make out why this definite integral is equal to 1. I graphed it using a...

Homework Statement [PLAIN]http://www4a.wolframalpha.com/Calculate/MSP/MSP184619c56eg84971107d00001e292h4340heh8g4?MSPStoreType=image/gif&s=35&w=135&h=53 It is a definite integral between x=0 and x=20. I forget how to integrate with a square root. Do you just change it to a power of a...

So, I was playing on Wolfram Alpha, and I managed to come up with this: http://www.wolframalpha.com/input/?i=integral_%28+infinity+*+sqrt%28-1%29+%29%5Epi+e%5E%28ix%29+dx&x=0&y=0 In Tex, I believe this is... \int_{i\infty}^{\pi}e^{i x} dx = i However, I have more than one problem...

Homework Statement The definite integral of (t^3 + t -1)/(sin(t)) from 2 to x^2 Homework Equations The Attempt at a Solution First off, I don't have the solution anywhere, my teacher just gave this to us to work on for the final exam review. I can think of a few things. I...

Homework Statement I get the answer 19/3 but the real answer is 343/3. Can you show work to how it is that? I did this many times and still came up with 19/3 instead I got up to evaluating the integral part of finding the area between two curves but can never get the correct answer. I...

Homework Statement Evaluate the definite integral for the area of the surface generated by revolving the curve about the x-axis. y=(x3/6) + (1/2x), [1,2] Homework Equations 2π∫[r(x)](1+[f'(x)2]) The Attempt at a Solution First I found the derivative. f'(x)= (x2/2) +...

Hello, what are the conditions that a continuous even function f must satisfy in order to have \int_{-\infty}^{+\infty}f(x)dx < \infty ?

Homework Statement What is wrong with the equation: \int^{\pi}_{0} sec^{2} x dx = tan x\right|^{\pi}_{0} = 0 Homework Equations None The Attempt at a Solution I don't know where to begin. I am inclined to give a mathematical (as opposed to a paragraph-like) explanation as to...

Homework Statement What is the definite integral of 1/(36+x^2) with bounds [0, 6]? I've only been taught U substitution to handle problems like these. I let u = 36+x^2 and du=2xdx but I am stuck and don't know what to do. Te answer is pi/24 but I don't know how to obtain it. Is there a...

Homework Statement \int^{1}_{-1} 2x\sqrt{(1-x^{2})^{3} +1} dx The Attempt at a Solution This is one of those questions in a math book that is almost never assigned and can be seen as "going above and beyond" what the chapter teaches you. I kind of improvised on the solution by substituting...

Homework Statement \int _{0}^{2\,\pi}\!\int _{0}^{1}\! 0.5\,{r}^{ 3.5}\sin \left( \theta \right) {\cos}^{2}+ 0.5\,{r}^{ 2.5}{\sin}^{2}+1/5\,{r}^{5}{\cos}^{2}{ dr}\,{d\theta} Homework Equations The Attempt at a Solution So I first integrate the r, which was relatively straight...

Orthonormal basis \big(\frac{1}{\sqrt{2}}, cos(2x)\big) \pi \int_{-\pi}^{\pi} sin^4(x)dx=\frac{3\pi}{4} Solve the integral without integrating. sin^4(x)=\big[\frac{1-cos(2x)}{2}\big]^2=\frac{1}{4} - \frac{2cos(2x)}{4} + \frac{cos^2(2x)}{4} I know I could rewrite cos^2(2x) but then I...

Homework Statement Hi guys, i have a exercise of the limit of a summation that is the formal definition of definite integral and i need resolve and explain, but i can't resolve for the rational exponent, for this, need help, thanks in advance. Homework Equations \lim_{n \rightarrow...

Homework Statement Need to evaluate int(-inf,inf)(x/(x^3+1)). Homework Equations The Attempt at a Solution I don't believe finding the residues will be a problem. However, the integral is over the whole real line, and there is a singularity at -1 on the real line, so I'm...

Homework Statement The letter W is defined as . The value of W, correct to the nearest hundredth, is? Homework Equations The Attempt at a Solution Now, I had this question on a test and got it wrong. I get 0.93, which I think is correct. Apparently the answer is 0.24. I call BS...

I'm studying a course in Fourier. In a multi-choice question, one of the answers asks for the value of the definite integral of sin(ax)/x over [-pi,pi]. I am wondering if there is a way to calculate this integral (I guess using Fourier techniques) or not. It is possible that it can't be solved...

Homework Statement This is am example from my textbook \int_{1}^{3} x^2 dx = 26/3 It then goes to define the arbitrary partition in [1,3] and maximum / minimum for each subinterval blah blah.. then it says For each index i, 1 <= i <= n, 3x_{i-1}^2 \leq x_{i-1}^2 + x_{i-1}x_i + x_i^2...

Homework Statement The Attempt at a Solution I have no idea how to start this integral. Can anybody give me a hint to start it off? Thank you.

This is not actually a homework problem. Rather, it is a problem from Courant and Robbins' What is Mathematics?, Chapter 8: "The Calculus", page 409-410. Homework Statement Prove that for any rational k =/= -1 the same limit formula, N → k+1, and therefore the result: ∫a to b xk dx =...