Questions about the evaluation of specific definite integrals.

learn more… | top users | synonyms (1)

2
votes
1answer
39 views

How do I integrate $\int_0^{2\pi} [x\sin x]\,dx $, where $[\cdot]$ is the greatest integer function? [on hold]

Integrate $$\int_0^{2\pi} [x\sin x]\,dx, $$where $[\cdot]$ is the greatest integer function.
1
vote
1answer
43 views

Evaluating $\int_0^\infty dn \, \frac{x^n}{(3n+1)(3n+2)}$

I'm trying to prove a particular series is convergent, and I would like to use the Cauchy integral test for fun, even though it's not the most convenient. I need to evaluate, $$\int_0^\infty dn \, ...
6
votes
0answers
55 views

Can we interchange the Integral and Summation when a limit is $\infty$?

I was trying to Evaluate the Integral: $$\Large{I=\int_1^{\infty} \frac{\ln x}{x^2+1} dx}$$ $$\color{#66f}{{\frac{1}{x^2+1} = \frac{1}{x^2\left(1+\frac{1}{x^2}\right)}=\frac{1}{x^2}\cdot ...
2
votes
5answers
85 views

Evaluating $\int_0^1 \frac {x^3}{\sqrt {4+x^2}}\,dx$

How do I evaluate the definite integral $$\int_0^1 \frac {x^3}{\sqrt {4+x^2}}\,dx ?$$ I used trig substitution, and then a u substitution for $\sec\theta$. I tried doing it and got an answer of: ...
0
votes
1answer
13 views

What part of area is included in the definite integral?

I am supposed to find the area of the blue shaded region. The line $\mathrm {OA}$ is $y=x$. The circle is $x^2+y^2=16$. The best method is to find slope of the line and use the formula ...
1
vote
2answers
18 views

Indefinite integrals with rati0nal and polynomial functions and Substituion

I am totally confused with the substitution method of evaluating indefinite integrals, especially those with rational functions and polynomials. I have 2 cases, which if I made to understand, would ...
7
votes
1answer
89 views

how to compute this limit

compute $I=\lim\limits_{n\to+\infty}\sqrt[n]{\int\limits_0^1x^{n+1}(1-x)\cdots(1-x^n)dx}$ attempt: i tried to evaluate the integral $$\begin{align} ...
1
vote
2answers
61 views

Solving $\int^{\pi}_{ 0.5\pi} \frac{dx}{1-\cos x}$

Evaluate $$\int^{\pi}_{ 0.5\pi} \frac{dx}{1-\cos x}$$ This is my attempt: $$\int^{\pi}_{ 0.5\pi} \frac{dx}{1-\cos x} = ...
1
vote
3answers
59 views

Evaluation of integral $\int_{-\infty}^{+\infty} xe^{-|x|}\,dx$ is not $0$

Given $$f(x)=\frac12e^{-|x|}, -\infty \le x \le +\infty$$ $$\int_{-\infty}^{+\infty} x f(x)\, dx= -\frac12\int_{-\infty}^{+\infty} x (-e^{-|x|})' dx=-\frac12\bigg(-xe^{-|x|} + ...
-1
votes
0answers
25 views

analytic solution of a definite integral

Integrate the following $$\int_0^\infty \alpha\,\beta\, c\, k\, x^s\, x^{c-1} (1+x^c)^{k-1} \left[(1+x^c)^k-1\right]^{-\beta-1} \left[1+\gamma ...
2
votes
3answers
47 views

Integral $\int_{0}^{\infty}\frac1{\sqrt[\alpha]{1+x^\beta}}dx$

Is there a general answer for the integral of the form:$$\int_{0}^{\infty}\dfrac1{\sqrt[\alpha]{1+x^\beta}}dx$$
0
votes
1answer
54 views

Integral $\int_0^\infty e^{-x/2}x\log(1+kx^2)\,dx$

How to evaluate: $$\int_0^\infty e^{-x/2}x\log(1+kx^2)\,dx$$ Basically am evaluating value of $\log(1+c\chi^2)$ where $\chi^2$ is $\chi$-squared distributed
1
vote
1answer
36 views

When does the integral converges?

For what $\alpha, \beta$ the integral $$\int_0^\frac{\pi}{2} \frac{(\frac{\pi}{2} - x)^\alpha}{(\cos x)^\beta} dx$$ converges? So first I've approved (using WolframAlpha) that $\frac{\pi}{2} - x ...
0
votes
0answers
10 views

Calculation of probabilities in Z table

I would like to calculate at least one probability from z table. I know that pdf for N(0,1) is 1/(2*pi)*exp^(-(x^2/2)). Also the cdf is However, I do not know how to calculate this integral. ...
3
votes
1answer
22 views

How to proceed with the following integration?

If $n$ is a positive integer, show that $$ \int_{\sqrt{n\pi}}^{\sqrt{(n+1)\pi}} \sin(t^2) dt = \frac{(-1)^n}{c}$$ for some $c \in [\sqrt{n\pi}, \sqrt{(n+1)\pi}]$ I have an idea that i can use Mean ...
1
vote
1answer
16 views

Prove that for every $T > \frac{\pi}{2} $, $\int_{\frac{\pi}{2}}^T \frac{cos(x)}{x}dx < 0$

I tried doing integration by parts a few times, after doing it 3 times I get the following expression: $$ \int_{\frac{\pi}{2}}^T \frac{cos(x)}{x}dx = \frac{sin(T)}{T} - \frac{1}{\frac{\pi}{2}} - ...
1
vote
3answers
38 views

How to convert into a definite integral

Could you show me how to convert the following into a definite integral: $$\lim\limits_{n \to \infty} \sum_{k=1}^{3n} \frac{1}{n}\cos\left(\frac{k\pi}{n}\right)\sin\left(\frac{2k\pi}{n}\right)$$ ...
0
votes
0answers
25 views

Simplify $\int_0^{\infty}\,dk\,\exp{(-\delta^2k^2)}\,J_1(kR)\frac{1}{1+bk^2}$

EDIT: I would love to find an analytical solution for this definite integral: $$\int_0^{\infty}\,dk\,\exp{(-\delta^2k^2)}\,J_1(kR)\frac{1}{1+bk^2}$$ with $\delta>0,\, R>0,\,b>0$. Does ...
1
vote
1answer
29 views

Compute integral given 2 other integrals

I want to know which solution is correct. The question states: If f is an integrable function on [1,3], and if $$\int_1^2f(x)dx=4 \space\space\space\space\space\space and \space\space\space\space ...
0
votes
1answer
33 views

How to compute $\sum\limits_{a=1}^{\infty}\int_0^b\lambda\left(\int_0^{\lambda}e^{-t}t^{a-1}dt\right)d\lambda$

Please suggest an efficient method to compute the following integral \begin{equation} I = \sum\limits_{a=1}^{\infty}\int_0^b\lambda\left(\int_0^{\lambda}e^{-t}t^{a-1}dt\right)d\lambda \end{equation} ...
1
vote
2answers
15 views

How to prove $L_{f}(P) \leq L_{f}(Q)$ when $Q$ and $P$ are partitions of $[a,b]$ and $Q \supseteq P$

I'm having trouble proving this idea. Suppose that $f$ is bounded on the interval $[a, b]$. $P$ and $Q$ are partitions of $[a, b]$, and $Q \supseteq P$. $$ L_{f}(P) \leq L_{f}(Q) $$ I know that this ...
1
vote
0answers
15 views

Evaluating the surface integral side of a divergence theorem problem

Edit: I realized where my mistake was...thanks for the help! In class we were discussing a problem (page 1185 in Smith and Minton's Calculus Third Edition): Let Q be the solid bounded by the ...
2
votes
2answers
70 views

Unusual integral

I got a clock as a gift recently. It has a very novel face in that the hour positions are given by a complex formula. For the most part, I have been able to verify the calculations presented as ...
1
vote
1answer
16 views

Volume of Revolution Verification

Question: A region in the $xy$-plane is bounded by the $x$-axis, the lines $x=1$, $x=2$ and the curve $y=2x^2 +1$. Find the volume obtained by rotating the region around the $x$-axis. I did ...
-4
votes
0answers
26 views

For the following integrals find a and find b [on hold]

In the following picture, what is a=? what is b=?
1
vote
1answer
39 views

Evaluation of the integral $I=\int_0^{\infty}[1-J_0(kr)]k^{1-\alpha}dk$

How to compute \begin{equation} I(r) = \int_0^{\infty}[1-J_0(kr)]k^{1-\alpha}dk,\quad 2<\alpha<4 \end{equation} where $J_0(x)$ is the zeroth order Bessel function of the first kind. A paper ...
1
vote
1answer
47 views

Convert to Riemann Sum

The following limit has to be converted to Riemann Sum. $$\lim_{N→∞}\sum^{N}_{n=-N}\left(\frac{1}{(N+in)}+\frac{1}{(N-in)}\right)$$ My attempt: ...
1
vote
1answer
39 views

Integral $\int_0^\infty \frac{e^{-cy} dy }{1+ay}$

$$I'=\int_0^\infty \frac{e^{-cy} dy }{1+ay}$$ a, b, and c, are positive coefficients. This integral is part of a problem which I'm trying to solve it and after lot's of effort the problem transform ...
1
vote
0answers
41 views

evaluating an indefinite and improper integral

$$ I=\int_0^{+\infty} \frac{Q_1(a,\sqrt{by})}{1+cy}dy $$ $a$, $b$, and $c$ are positive coefficients. $Q_1$ is Marcum $q$-function. This integral is part of a problem which I'm trying to solve it and ...
0
votes
1answer
58 views

Estimating the integral $\int \frac{\sin(x)}{x}\, dx$. [on hold]

Would anyone be able to help me out with this question? I'm not quite sure how to go about it. Thanks in advance! Consider the integral $$ I = \int_{\pi/2}^\pi \frac{\sin x}{x}\,dx. $$ This integral ...
10
votes
4answers
157 views

How to compute $\int_0^{\infty} \frac{\sqrt{x}}{x^2-1}\mathrm dx$

Could you explain to me, with details, how to compute this integral, find its principal value? $$\int_0^{\infty} \frac{\sqrt{x}}{x^2-1}\mathrm dx$$ $f(z) =\frac{\sqrt{z}}{z^2-1} = \frac{z}{z^{1/2} ...
0
votes
0answers
33 views

Riemann sum/integral of $\sin(x)$ from $0$ to $A$ [duplicate]

Hello I keep getting stuck on calculating the Riemann sum/integral of $\sin x$ from $0$ to $A$ I know this has been looked at before but I just don't understand it and was hoping someone could ...
0
votes
1answer
32 views

Riemann Integral Property for Continuous, Monotonic, Non-negative Function

If $f$ is continuous, non-negative, and monotonically increasing function on $[0,∞)$, then prove that $\int^{x}_{0} f(t)dt\leq xf(x)$ $\forall x ≥ 0$ My attempt: Define $F(x)=\int^{x}_{0} f(t)dt$. ...
-2
votes
0answers
20 views

Definite integrals and piecewise defined functions [on hold]

Consider the function $G(x) = \int_0^x g(u)\, du$ , where: $ g(u) = \begin{cases} 4 - \frac 43 u, & \text{for $0 \leq u < 6$} \\ u - 10, & \text{for $6 \leq u \leq 12$}. \end{cases} $ i. ...
5
votes
1answer
183 views

Why do we put absolute brackets for ln?

When writing out the final answer in $\ln$ form, why is it necessary to put absolute brackets? How does it affect the answer? I have this answer of $-3\ln|\frac{3+\sqrt{9-x^2}}{x}|$, but why does it ...
2
votes
2answers
46 views

what is the integral $\int_{-\pi/3}^{\pi/3} \tan (\theta)$?

I tried evaluating the integral $\int_{-\pi/3}^{\pi/3} \tan (\theta)$. I keep getting $\ln(2) - \ln(2) = 0$, but my textbook says its $\ln(4)$. I'm not sure what I am doing wrong because when I ...
0
votes
3answers
30 views

How does integrating over absolute values work with definite integrals?

I have $ \int_0^\pi | \sin(x/2) | \, dx $, and according to Wolfram Alpha, the indefinite integral is: $$ -2\cos(x/2)\operatorname{sgn}(\sin(x/2)) + C $$ but the definite integral above evaluates to ...
0
votes
1answer
38 views

$\frac{1}{2} \int_{a}^{b} f = \int_{a}^{c} f$ [on hold]

$f$ an integratable function defined in $[a;b] \rightarrow \mathbb{R}$: prove that exists $c \in [a;b]$ that: $\frac{1}{2} \int_{a}^{b} f = \int_{a}^{c} f$ and then give an example that might not ...
1
vote
1answer
48 views

How to evaluate $\lim_{n\to\infty}\int_0^n\frac{x^2+a^2}{x^4+b^2x^2+b^4}dx$

Evaluate this limit: $$\lim_{n\to\infty}\int_0^n\dfrac{x^2+a^2}{x^4+b^2x^2+b^4}dx$$ I tried to simplify this fraction. I noticed that $x^4+b^2x^2+b^4$ can be written as $$\dfrac{x^6-b^6}{x^2-b^2}$$ ...
1
vote
3answers
55 views

Evaluate $\lim_{t \rightarrow 0} \int_{0}^t \frac{1}{f(u)}du$

Let $f(u)$ be a function such that $\lim_{u \rightarrow 0} f(u)=0$ e.g. $f(u)={u}$. How would I evaluate $$ \lim_{t \rightarrow 0} \int_{0}^t \frac{1}{f(u)}du $$ Is this always equal to zero? My ...
3
votes
2answers
117 views

Find $\int_0^a{f(x)}\, dx$

SMT 2013 Calculus #8: The function $f(x)$ is defined for all $x\ge 0$ and is always nonnegative. It has the additional property that if any line is drawn from the origin with any positive slope $m$, ...
-1
votes
0answers
17 views

Need help to solve double integral exercise

I'm facing problems solving these integrals. I can't reach the result. Could anyone help me? There are two problems with the same integral. Integral: $\iint (y) dx dy $, a) $\{B=(x,y) \in R² | ...
0
votes
3answers
42 views

integrals calculation got wrong with the extra 2

Given $$ f(x, y) = \begin{cases} 2e^{-(x+2y)}, & x>0, y>0 \\ 0, &otherwise \end{cases}$$ For $ D: 0 <x \le 1, 0 <y \le2$, I'm trying to calculate this $$ \iint_D f(x,y) \, dxdy ...
0
votes
1answer
51 views

$\int_0^{\pi/2}\ln(\sin(x))$?

From this paper: http://math.ucsd.edu/~ebender/20B/7_DefInt.pdf Shouldn't $du$ be $dt$? And also how do you get from that line to the final result if $du$ is not $dt$?
2
votes
7answers
243 views

Why it's true? $\arcsin(x) +\arccos(x) = \frac{\pi}{2}$

The following identity is true for any given $x \in [-1,1]$: $$\arcsin(x) + \arccos(x) = \frac{\pi}{2}$$ But I don't know how to explain it. I understand that the derivative of the equation is a ...
3
votes
2answers
39 views

Real integral getting numerically evalued to complex number

I tried feeding th following integral into WolframAlpha: $$\int_0^2\frac{1}{3\sqrt{x}(\log(x))^{\frac13}},$$ to get an idea of its value. Result: http://bit.ly/strangeint ...
0
votes
1answer
20 views

Finding hypervolume lying between Gaussian function and x-y-z plane over $\mathbb{R}^3$

Define the 3-variable Gaussian function by $G(x,y,z) = e^{-(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2})}$. Find the hypervolume lying between this surface and the x-y-z hyperplane, over the ...
1
vote
1answer
26 views

Simplification of integration region. (Shuffle product?)

Let us define the usual $n$-dimensional simplex: $$\Delta_{a,b}^n = \{x_1,\dots, x_n\in [0,1]^n: a<x_1<\cdots <x_n<b\}.$$ Imagine we have an integral like: $$I:=\int_{\Delta_{a,b}^n} ...
0
votes
1answer
20 views

Integral containing Associated Legendre Polynomials

I need to evaluated the following integral: $\int_0^\pi \sin(x) \cos(x) P_l^m(\cos x) P_k^m(\cos x) \mathrm{d}x$ and I thought since a solution is known to a similar thing $\int_0^\pi \sin(x) ...
0
votes
1answer
29 views

Integral of nonnegative function on plane domain gives a negative result, what is wrong?

Given an area $D: x \ge y, 0 \le x \le 1, y \ge 0$. $$ f(x,y)= \begin{cases} 2, & (x,y) \in D,\\ 0, & \text {others}\end{cases} $$ For this area $D_1: x+y \le 1, 0 \le y \le x$, I'm ...