Definite integral Definition and 391 Threads

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...

$\displaystyle \int\frac{1}{(3+4\sin x)^2}dx$

π/3 ∫tan^5 x sec x 0 I want to split the tan^5 x into a (sec^2 x - 1) but I only have sec x. So I'm a bit stuck. Help is appreciated.

How to evaluate the definite integral!? Homework Statement Integral test to see if the series converges or diverges. (infinity) Sigma ((n^2)/(e^(n/3)) (n=1) Homework Equations First I derive to make sure the function is decreasing and positive so it fits the...

Homework Statement If \vec{r}(t) = t^2\vec{i} + t\cos{(\pi t)}\vec{j} + \sin{(\pi t)}\vec{k}, evaluate \int_{0}^{1} \vec{r}(t) \text{dt}. Homework Equations The Attempt at a Solution So I tried integrating each individual part, and I got \frac{1}{3}t^3\vec{i} + (-\pi t\sin{(\pi t)} -...

Hello, I am going through Whittaker's treatise on Classical Mechanics. In chapter 3 he derives the equation of motion for a simple pendulum, and I have a question about his method. Starting from the general form for the equation of energy (s is the path): \frac{m}{2}\dot{s}^2 =...

Homework Statement ∫ln(x+1)/(x+1)2dx from a=1, b=2 (Answer is -ln3/3+ln2/2+1/6) Homework Equations ∫fdg=fg-∫gdf The Attempt at a Solution Let u = (x+1), du=dx ∫lnu/u2du Let f=lnu , df = 1/udu, g=-1/u, dg=1/u2du =-lnu/u+∫1/u2du =[-lnu+1/u] 1->2 Using Fundamental Theorem...

If f(x)=\int_0^\infty g(x) dx and I wanted to multiply the integral by, say, a, what would I multiply the left side by? In other words, ? \times f(x) = a \int_0^\infty g(x) dx Thanks in advance!

Homework Statement Definite integral, from 0 to 3, of Square root of 1+t^3.Homework Equations Tried substitution The Attempt at a Solution There should be a simple way to do this but I can't seem to figure it out. Tried the substitution and whatnot but couldn't reach an answer.

Hi, I've been trying to evaluate the following integral \int_{z}^{\infty}\exp\left(-y^{2}\right)\mathrm{erf}\left(b\left(y-c\right)\right)\,\mathrm{d}y or equivalently \int_{z}^{\infty}\exp\left(-y^{2}\right)\mathrm{erfc}\left(b\left(y-c\right)\right)\,\mathrm{d}y...

Homework Statement This problem was taken from Spivak's Calculus 3rd Edition, Problem 19-25 (d). Let f be integrable on [-1 , 1] and continuous at 0. Show that \lim_{h\rightarrow 0^+} {\int_{-1}^{1}{\frac{h} {h^2+x^2}}f(x)dx = {\pi}f(0).} Homework Equations I already proved from...

Homework Statement I'm working on a long problem and have come to the final step. The answer seems so simple, but I can't quite get to it. I need to evaluate this integral: \int_0^{\infty}\ \left(e^{-k^2 t}/k\right)\sin(kx)\ dk Homework Equations Mathematica gives the result as...

Dear All, \int_{0}^{1}x*\left(\left(-ln(x)\right)^{k}\right)*\left(1-x\right)^{(N-1)}dx , where k is an odd no. N >=2.regards, Bincy

Hii friends,\int_{0}^{1}\left(1-x\right)^{N}*e^{x}\, dx regards, Bincy

Homework Statement Find the value of ∫ ( sin 37x / sin x )dx - from 0 to π Homework Equations The Attempt at a Solution Well i tried doing it by parts but it makes its more complicated. I tried to expand sin 37x as sin(36+1)x but could not solve it. I am lost

Hi everyone ... I am a first time poster from South Africa. I have been visiting the forum for some time. I am busy teaching myself calculus and physics. I have hiccup with the concept of definite integral and arc length of a function. In my understanding these should be the same thing...

Homework Statement ∫[|2x-1|] evaluated on the interval [1,-1] Homework Equations The Attempt at a Solution I know we must split it into two equations ∫ from [a,c] + ∫ from [c,b] and in absolute value one is negative and one is positive, so it will be 2x-1, and 1-2x my...

Hi. Homework Statement I'm solving the definite integral 4π∫{0,3/2}(x^3)dx For some reason I get a different answer solving by hand and using my TI-89 than I do when I use an online calculator or my TI-36X Pro. I have to use the TI-36X Pro to take my Engineering FE exam in 2 & 1/2...

Find the limit "a" such that the definite integral equals 8 Homework Statement Hello! the problem ask me to find for which a in R, does this holds: 2 ∫ abs(2x - x^2) = 8 a The Attempt at a Solution Well, I see that: f(x) { if x > 2, then f(x) is negative; if x < 2, then f(x) is positive...

Homework Statement Find a power series expansion about x = 0 for the function f(x) = ^{1}_{0}\int\frac{1 - e^{-sx}}{s} ds Homework Equations The power series expansion for a function comes of the form f(x) = ^{\infty}_{0}\sum a_{k}x^{k} The Attempt at a Solution I've tried...

the definite integral is: integral from 0 to 1 of ln(x). i used integration by parts (u=ln(x), du=1/x dx, dv=dx, v=x) to show that the integral is equal to: [x*ln(x)] (1,0) - integral from 0 to 1 of dx. this gives 1*ln(1)-0*ln(0)-(1-0) ln(1)=0, so the equation is now 0*ln(0)-1 0*ln(0) is an...

Hi everyone, when I was dealing with a physics problem, I find a definite integral and I cannot solve it. \int_{0}^{2\pi}\frac{1}{(a+cosx)^{2}}dx a is constant. Thank you!

Show that $\displaystyle \int_{0}^{\infty}\frac{x^{\alpha-1}}{(w+x^{\beta})^{\gamma}}\ dx = \frac{w^{\frac{\alpha}{\beta}-\gamma}}{\beta} B \left( \frac{\alpha}{\beta}, \gamma - \frac{\alpha}{\beta} \right) = \frac{w^{\frac{\alpha}{\beta}-\gamma}}{\beta} \frac{\Gamma \left(...

problem: sin(x)^2cos(x) integrate from 0 to pi/2the way I tried this is by making u = sin, and then working with du to make cos cancel out the next thing I did was check my new intervals by plugging in pi/2 and 0 into the u function (becomes 0 to 1) next, I took the anti derivative of u, which...

Homework Statement make a u- substitution and integrate from u(a) to u(b) Homework Equations ∫[0,1] √(t^5+2t) (5t^4+2) dt The Attempt at a Solution u= t^5+2t du= 5t^4+2 u(1)=2 u(0)=0 ∫[0,1] √(u) du (2/3)(u)^3/2+c l[0,2] (2/3)( 0^5+2(0))^3/2- (2/3)(2^5+2(2))^3/2 0+...

Homework Statement ∫sin2xcosx x=0,pi/4 Homework Equations The Attempt at a Solution By double angle magic: ∫sin2xcosx= ∫2sinxcos^2x u=cosx du=-sinxdx -2∫u^2 -2(u^3)/3 -2cos^3x/3 = -1/(3√2) for x=pi/4 How does that look folks?

Homework Statement I'm trying to evaluate \int_0^{2\pi} \frac{9-6\cos t}{5-4\cos t} dt \ . I found that the antiderivative of the integrand is F(x) = \frac{3}{2}x + \tan^{-1} \left(3\tan\frac{x}{2} \right) so using the FTC, the integral should be F(2\pi)-F(0)=3\pi But using a...

Proof of Definite Integral?? Riemann Sum helps us to find the are under a curve.But I couldn't understand connection between Riemann Sum and Definite Integral.I mean; \int^{b}_{a}f(x).dx=F(b)-F(a) How can I prove F(b)-F(a) in definite integral?? Thanks..

Homework Statement The Attempt at a Solution I don't see why the √v disappears in this step I understand how they got the 20/3, because they integrated, but if they're integrating then I would think v^-1/2 would become (v^1/2)/(1/2)

Homework Statement Why does the 3x^2 disappear? Doesn't it play a role in the answer?

Homework Statement If n is a positive integer, then \lim_{n\to\infty}\frac{1}{n}\left[\left(\frac{1}{n}\right)^2+\left(\frac{2}{n}\right)^2+\cdot\cdot\cdot+\left(\frac{n-1}{n}\right)^2\right]can be expressed by what definite integral? The attempt at a solution A student I was helping...

Homework Statement Integrate from 0 to 1 x47e-x48 Homework Equations The Attempt at a Solution u = -x48 du = -48x47dx dx = du / -48x47 ∫eu * du/-48 -\frac{1}{48} ∫ eu du -\frac{1}{48} eu -\frac{1}{48} e1 - ( -\frac{1}{48} e0) -\frac{e}{48} +...

Can you solve the Integral? \[ I(a,b)=\int_{0}^{\pi / 2}\frac{1}{(a\cdot\cos^2{x}+b\cdot\sin^2{x})^2}dx \quad a,b>0 \]

Homework Statement "Integrate both sides of these equations from some starting point to some ending point (such as t0 to t): r dt = d[A], C dt = dH and dw = (nRT/V) dV Homework Equations -- The Attempt at a Solution I'm have trouble understanding the concept of taking the...

Homework Statement \int\stackrel{tan(8)}{tan(1)}\frac{dx}{x^{2}+1} Homework Equations The Attempt at a Solution \int\stackrel{tan(8)}{tan(1)}\frac{dx}{x^{2}+1}tan^{-1}x|\stackrel{tan(8)}{tan(1)}tan^{-1}(tan(8))-tan^{-1}(tan(1))8-1=7I'm fine with this, but my instructor said "Hint: The answer...

1. Find the expected value of the largest order statistic in a random sample of size 4 from the standard normal distribution. 2. Homework Equations E(X(4,4))=4∫xf(x)(F(x))^3dx, (from minus infinite to plus infinite), where f(x) is the probability density function of standard normal...

We are just starting integrals right now and I'm having trouble with this problem. I know how to solve a definite integral but I don't quite understand what is being asked here? Just wondering if anyone could let me know where I should start? if∫0,3f(x)dx=12, ∫0,6f(x)dx=42,find...

Homework Statement Use the form of the definition of the integral to evaluate the following: lim (n \rightarrow ∞) \sum^{n}_{i=1} x_{i}\cdotln(x_{i}^{2} + 1)Δx on the interval [2, 6] Homework Equations x_{i} = 2 + \frac{4}{n}i Δx = \frac{4}{n} Ʃ^{n}_{i=1}i^{2} =...

∫cos x + sin x lower limit: 0 upper limit: pi correct answer: 4 intervals: 1.8962, 8 intervals: 1.9742, 16 intervals: 1.9936 i cannot seem to get the correct answer, here is my attempt: x values cosx + sinx values 0...

Homework Statement Working my way through Zienkiewicz's "FEM its basis & fundamentals" I am stuck on the following example: - \int_{y_1}^{y_2} t N1 \left\{ \begin{array}{c} \sigma_x \\ \tau_y \end{array} \right\} dy = -\left\{ \begin{array}{c} k_x t (2 y_1 + y_2) (y_2 - y_1) / 6 \\ 0...

Homework Statement I've been solving a problem, the solution is complete, however, I must prove that the following relation is true: \int_{0}^{\frac{\pi}{2}} sin^m 2x dx = \int_{0}^{\frac{\pi}{2}} cos^m x dx for any m. Homework Equations - The Attempt at a Solution Well, I've trying to...

Homework Statement integral from 0 to pi of (x sinx)/(1+cos^2x)Homework Equations I just need to know if I did it rightThe Attempt at a Solution I got (5pi/6)-1/9

Homework Statement Evaluate the limit by expressing it as a definite interval integral \frac{lim}{n \rightarrow \infty} (\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{n+(n-1)}+\frac{1}{n+n} Homework Equations The Attempt at a Solution OK, so I know it has to do with Rieman sums...

I'm reviewing my Calc 1 material for better understanding. So, I was reading about the area under a curve and approximating it using Riemann sums. Now, I understand the method, but I was a little confused by finding xi*. I know there is a formula for it xi*=a+Δx(i). What does the "i" stand for...

Homework Statement (2(x)^2)/((x+1)((x)^2+1)) from 0 to 1 The indefinite answer is (1/2)ln2+ln2-(pi/4) How did it get to this answer?(3/2)ln2-(pi/4)

so the question is how to solve the next: ∫...∫exp(-(a+b*x1+c*x1*x4+d*x3*x4+f*x2*x3*x4)^2)dx1dx2dx3dx4 a,b,c,d,f - real constants; x1,x2,x3,x4 - real variables; borders of integration are finite, let say for x1 it is [x1a,x1b] and so on for the rest of them; i'v tried to solve it in...

Homework Statement Considering the appropriate complex integral along a semi-circular contour on the upper half plan of z, show that \int^{\infty}_{\infty} \frac{cos(ax)}{x^2 + b^2} dx = \frac{\pi}{b}e^{-ab} (a>0, b>0) Homework Equations \int_{C} = 0 For C is a semi-circle of...

I don't understand how to evaluate a fraction in a definite integral, could anyone please explain this problem to me? int. between 3 (lower integral) and 4 (upper integral) of 3x^2+2/x^2 I thought of using the substitution law, and got as far as 1/6 u * du/x ...I don't quite know how to...

Homework Statement \int_{1/2}^{\sqrt{3}/2}\frac{6}{\sqrt{1-t^2}}dt Homework Equations The Attempt at a Solution Ok, this is an odd problem I'm working from the book. The book says the answer is \pi. First I tried getting rid of the fraction: \frac{6}{\sqrt{1-t^2}}dt =...

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...