Questions about the evaluation of specific definite integrals.

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2
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2answers
36 views

Why does this sum equal zero?

Le}t $\gamma$ be a piece-wise, smooth, closed curve. Let $[t_{j+1}, t_{j}]$ be an interval on the curve. Prove, $$\int_{\gamma} z^m dz=0$$ In the proof it states $$\int_{t_{j}}^{t_{j+1}} ...
0
votes
0answers
12 views

Showing certain sum as a Riemann-Stieltjes integral

Let $e(\beta) = e^{2 \pi i \beta}$. I am reading an article, where the author defines the following sum $$ S(N) = \sum_{0 \leq x \leq N, x \equiv g (mod \ q)} \Lambda(x) e(f(x) \alpha), $$ where $f$ ...
6
votes
3answers
69 views

How do I evaluate this : $\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ dx $ for $a > 0$?

How do I evaluate this integral if I supposed that : $a > 0$ $$\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ dx .$$ For $a=2$ I have got : $2\pi$ I think the result will be : ...
1
vote
1answer
17 views

Error bound of midpoint rules with unbounded second derivative

It is well known that error bound of midpoint rule for function $f[a,b]$ is given by $$ E\leq K\frac{(b-a)^3}{24 n^2} $$ where $|f(x)''\leq K|$ and $n$ is the number of time steps. if second ...
4
votes
1answer
65 views

Area under tangent to a curve.

The tangent to the graph of the function $y=f(x)$ at the point with abscissa $x=a$ forms with the line $x$-axis an angle $\frac{\pi}{6}$ and at the point with abscissa $x=b$ an angle of ...
-1
votes
0answers
39 views

How to find the following definite and indefinite integrals [duplicate]

I want to calculate the integral $$\int_0^{\pi \over 2}e^{ \sin t}dt$$ can we find a primary function for $f(t) = e^{\sin t}.$ With many thanks.
1
vote
2answers
39 views

Finding the following definite and indefinite integrals

I want to calculate the integral $$\int_0^{\frac{\pi}{2}} e ^{ \sin t}\, dt.$$ Can we find a primitive function for $f(t) = e ^{\sin t}$?
2
votes
2answers
58 views

Finding $\int_{0}^{\frac{\pi}{2}}\frac{1}{\cos (x-\frac{\pi}{3}).\cos (x-\frac{\pi}{6})}\mathrm{d}x$

How can I find $$\int_{0}^{\frac{\pi}{2}}\frac{1}{\cos (x-\frac{\pi}{3}).\cos (x-\frac{\pi}{6})}\mathrm{d}x$$ ? I suspect this has something simple to do with the basic definite integral properties; ...
-3
votes
2answers
45 views

How to calculate the integral?? [on hold]

I want to calculate the integral $$\int_0^{\pi \over 2}e^{c \sin^2t}dt$$ for a constant parameter $c \in \Bbb R.$ With many thanks.
2
votes
1answer
56 views

Find the area bounded between $f(x)=\frac{\arctan(x)}{x^2}$ and $g(x)=\frac{\arctan(x)}{x^2+1}$

Find the area bounded between $$f(x)=\frac{\arctan(x)}{x^2} \quad\text{and}\quad g(x)=\frac{\arctan(x)}{x^2+1}.$$ The title says the question. The limits are from 1 to infinity. I know that I ...
6
votes
0answers
44 views

Integral of exponencial

Calculate the following integral $$\int_0^{+\infty} \exp\left(-a^2 x\left(\dfrac{x-6}{x-2}\right)^2\right) \dfrac{dx}{\sqrt{x}}$$ I think the relation between this integral and function gamma is ...
1
vote
1answer
16 views

Evaluating the volume of a torus formed by rotating a region about a horizontal axis using shells.

Using the method of cylindrical shells, find the volume of the shape created by revolving the region $x^2+(y-5)^2=4$ about $y=-1$. A cylindrical shell is given by: $2\pi v f(v) \ dv$ I solve ...
12
votes
1answer
178 views

Calculating 2 integrals in polylogarithmic functions

Are we aware of any nice way of calculating these $2$ integrals? $$i) \space \int_0^1 \frac{\text{Li}_2\left(x-x^2\right)}{x^2-x+1} \, dx$$ $$ii)\space \int_0^1 ...
2
votes
2answers
25 views

$F=w(x)=\frac{k}{x^2}$ How much work required to lift a satellite an “infinite distance” into outer space?

The satellite is 6000 lbs at earth's surface, a distance $R$ from the earth's centre (so the answer will be in terms of $R$). I know that it's supposed to be an improper integral going from 0 (?) to ...
0
votes
1answer
14 views

Evaluate the volume of a solid of revolution using shells.

A cylindrical shell $S$ formed by some revolution about the $y$-axis is given by the equation: $S=2\pi x f(x)dx$, where the circumference $C$ of the shell is $C=2\pi x$, the height of the shell ($H$) ...
0
votes
1answer
54 views

About odd functions and improper integrals e.g. $\int^{\infty}_{-\infty}\sin x \; dx$

Does $\displaystyle \int^{\infty}_{-\infty}\sin x \; dx$ converge? Since $\sin x$ is an odd function, and we know that in definite integrals $\displaystyle \int^{a}_{-a}\sin x \; dx=0$ then does ...
3
votes
2answers
155 views

Evaluate an Integral

Evaluate: $$\int_{0}^{\infty}\dfrac{\sin^3(x-\frac{1}{x} )^5}{x^3} dx$$ I've been stumped by this Integral and cannot think of how to evaluate it. I substituted $\dfrac{1}{x^2}=t \Rightarrow ...
0
votes
2answers
49 views

Integral with only a list of values

I am supposed to perform an integral of function $y(x)$ from say $x_1$ to $x_2$. Now the issue is I don't have an actual function $y(x)$, but I do have a list of values for $y$ and $x$. In what way ...
1
vote
2answers
24 views

Definite integral application

So, the task is to calculate the area of a shape in xOy plane bounded by functions: $y = x\sqrt{4x-x^2}$ and $y = \sqrt{4x-x^2}$ Could you please explain how I can solve this? How can I find the ...
1
vote
0answers
26 views

Evaluate $\int_0^1 \frac{P_3(t)}{\sqrt{1-k^2 P_3^2(t)}}dt$

We have the following integral: $$\int_0^1 \frac{P_3(t)}{\sqrt{1-k^2 P_3^2(t)}}dt$$ where $P_3(t)$ is a third-degree polynomial with all coefficients different from zero. Is it an elliptic integral? ...
3
votes
1answer
148 views

What is the minimum degree of a polynomial for it to satisfy the following conditions?

This is the first part of a problem in the high-school exit exam of this year, in Italy. The differentiable function $y=f(x)$ has, for $x\in[-3,3]$, the graph $\Gamma$ below: $\Gamma$ exhibits ...
4
votes
3answers
124 views

About the integral $\int_{0}^{1}\frac{\log(x)\log^2(1+x)}{x}\,dx$

I came across the following Integral and have been completely stumped by it. $$\large\int_{0}^{1}\dfrac{\log(x)\log^2(1+x)}{x}dx$$ I'm extremely sorry, but the only thing I noticed was that the ...
1
vote
1answer
18 views

Integral Equation Unknown Limits

What is the name of an equation, where the unknown is one of the limits of integration? Is there a theory that studies such equations, standard methods of solution? The simplest example is the ...
6
votes
6answers
262 views

Compute definite integral

Question: Compute $$\int_0^1 \frac{\sqrt{x-x^2}}{x+2}dx.$$ Attempt: I've tried various substitutions with no success. Looked for a possible contour integration by converting this into a rational ...
0
votes
0answers
36 views

Can I have variables extreme of integration?

Suppose you have a function $v(t)$ that you want to find. The condition is that it's integral is some fixed quantity. The integral is done between $0$ and $u(t)$, where $u(t)$ is an increasing ...
1
vote
0answers
15 views

Arclength of intersection between 2 perpendicular cylinders.

hi have 2 perpendicular cylinders that intersect (I read the resulting curve is called the Steinmetz curve). $x^2+y^2=R_1^2$ and $y^2+z^2=R_2^2$, with $R_2\lt R_1$ and want to parametrize the length ...
1
vote
0answers
68 views

Make this integral zero

Consider the integral $$\int_0^\infty x^nf(x)\,\mathrm{d}x$$ from this answer. The integral is zero for the following $n$ and $f(x)$: $n=4k$, $f(x)=e^{-x}x^{-1}\sin(x)$ $n=4k+1$, ...
0
votes
1answer
25 views

Integration problem of a modified 'standard integral'

Consider $\int\limits_0^\infty ye^{-y}e^{-xy}dy$ I can use the fact that $\int\limits_0^\infty u^ne^{-u}=n!$ Clearly, $\int\limits_0^\infty ye^{-y}e^{-xy}dy=\int\limits_0^\infty ye^{-y(1+x)}dy$. ...
4
votes
1answer
41 views

Simplifying real part of hypergeometric function with complex parameters

I am looking for a simpler representation of the following hypergeometric function with complex parameters in terms of more basic functions and manifestly real parameters: ...
4
votes
3answers
63 views

Help with a Series (Edited)

The original problem was: $$\sum_{k=0}^\infty\dfrac{k}{6k^3+13k^2+9k+2}$$ Using Partial Fractions, I resolved this into ...
3
votes
2answers
68 views

How to prove Raabe's Formula [duplicate]

For quite some time, I've been trying to prove Raabe's Formula, or in other words: $$\int_a^{a+1} \ln\bigg(\Gamma(t)\bigg)dt=\dfrac{1}{2}\ln(2\pi)+a\ln(a)-a$$ This is how I tried: ...
0
votes
2answers
47 views

Finding the derivative of an integral with variable limits: ${\mathrm{d} \over \mathrm{d}x}\int_{x}^{x^2}{1 \over -2y}e^{-5xy^{2}}\mathrm{d}y$?

How do you compute the derivative $${\mathrm{d} \over \mathrm{d}x}\int_{x}^{x^2}{1 \over -2y}e^{-5xy^{2}}\mathrm{d}y$$ where the integral has variable limits?
2
votes
1answer
75 views

Hint to integrate $\int_0^1 \frac{\log(1+x)}{x} \, \mathrm{d}x$?

How to integrate $$\int_0^1 \frac{\log(1+x)}{x} \, \mathrm{d}x\text{ ?}$$ The answer is $\frac{\pi^2}{12}$, but I don't seem to get a way to reach there.
1
vote
1answer
64 views

Multiple Integrals

$$\int _{ 5 }^{ 20 }{ \int _{ 5 }^{ 20 }{ \int _{ 5 }^{ 20 }{ \int _{ 5 }^{ 20 }{ \ln(w+x+y+z) }\ dw\; dx\; dy\; dz } } }$$ Unfortunately I cannot think of how to approach this problem. The only ...
6
votes
1answer
109 views

How to compute the integral $ I\left(c\right)=\int_{0}^{1}{\frac{\ln(1-cx)}{1+x}dx} $

I am currently working on this question and the following integral came up: $$ I\left(c\right)=\int_{0}^{1}{\frac{\ln(1-cx)}{1+x}dx} $$ for a suitable c. I would like to compute it in terms of ...
-2
votes
1answer
52 views

Does the integral $\int_{0}^{1} \frac{e^{-x}}{x}dx$ converge? [closed]

This problem was given in my Calculus 1 class. I tried to find the anti-derivative so I could evaluate it but I didn't have any luck. How should I proceed? Thanks.
7
votes
5answers
336 views

Definite integral of even powers of Cosine.

I'm looking for a step-by-step solution to the following integral, in terms of n$$\int_0^{\frac{\pi}{2}} \cos^{2n}(x) \ {dx}$$I actually KNOW that the solution is$${\frac{\pi}{2}} \prod_{k=1}^n ...
4
votes
3answers
101 views

Finding $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{1}{1+\sqrt{\tan x}}dx$ [duplicate]

How can I integrate $$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{1}{1+\sqrt{\tan x}}dx\ \ \ ?$$ I have made the integral into the form of $\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}$, but ...
3
votes
1answer
110 views

Evaluate $\int_0^1 \frac{\ln \left(a+\sqrt{a^2+1}\right)}{a\sqrt{a^2+1}}da$

Problem: Evaluate$$\int_0^1 \dfrac{\ln \left(a+\sqrt{a^2+1}\right)}{a\sqrt{a^2+1}}da$$ Unfortunately I have absolutely no idea as to how to approach this problem. It was suggested that I try ...
2
votes
2answers
68 views

Help with the integral $\int_{0}^{\infty}\frac{y^{2}e^{y}}{e^{sy}+e^{-sy}-2}dy$

I want to do the integral : $$I(s)=\int_{0}^{\infty}\frac{y^{2}e^{y}}{e^{sy}+e^{-sy}-2} \, \mathrm{d}y$$ $s$ being a complex parameter. I tried expanding the dominator of the integrand, but this way ...
2
votes
0answers
33 views

Smart coordinates for six-dimensional integral

I have a (hopefully) simple question: I am dealing with a definite (on all of $\mathbb{R}^6$) six-dimensional integral $$\int_{\mathbb{R}^6} F(\vec{x}_1,\vec{x}_2)d^3x_1d^3x_2$$ where the function ...
1
vote
2answers
38 views

Evaluate: $\dfrac{u^{2n+1}\ln(u)}{2n+1}\bigg|^{u=1}_{u=0}$ at $u=0$

Problem: Evaluate: $$\dfrac{u^{2n+1}\ln(u)}{2n+1}\bigg|^{u=1}_{u=0}$$ This actually came up when I tried to solve the Integral $$\int_0^1 u^{2n}\ln(u) du$$ by using IBP. The problem is that I am ...
0
votes
1answer
30 views

Is $ y = \int_0^x \frac{du}{\sqrt{x^4 - u^4}} $ an increasing or decreasing function of $x$? Find an integral expression of $y'$.

Is $$ y = \int_0^x \frac{du}{\sqrt{x^4 - u^4}} $$ an increasing or decreasing function of $x$? Find an integral expression of $y'$. I don't understand what I need to know to solve this question. It ...
0
votes
1answer
54 views

Finding the area between $x \sqrt{4x-x^2}$ and $\sqrt{4x-x^2}$

So I've been doing real analysis for a last couple of days, and stumbled upon this task. The task is to find the area enclosed by $$y_1=x\sqrt{4x-x^2} $$ and $$y_2= \sqrt{4x-x^2} $$ This is one of ...
0
votes
1answer
62 views

A trig ratio integral

Consider the two integrals, where $a$ and $n$ are integers, \begin{align} I_{1} &= \int_{-n \pi}^{n \pi} \frac{\tan^{2a}x \, dx}{\tan^{2a}x + \cot^{2a}x} \\ I_{2} &= ...
0
votes
2answers
88 views

Limit of definite integral divided by $x^2$ - no idea how to solve it

I have to solve a limit of a definite integral, but honestly I have absolutely no idea how to do it. I solved this integral (I just solved $\int \frac{1}{t}$, $-\int \ln(1+t)$, and $\int \ln(t)$) but ...
0
votes
0answers
31 views

Is there a better way to determine the function in the integrand?

I need to find $U(z)$ given that $\Delta\ll 1$. $$\int_{-\Delta/2}^{\Delta/2} U(z) \, dz = C$$ $C$ and $\Delta$ are constants. Since $\Delta$ is small I am just using $$ U(z) = C / \Delta\,.$$ It ...
2
votes
0answers
31 views

Is this type of definite integral defined?

I was trying to generalise the definite integral in the following way: Let $f:[a,b] \to \mathbb{R}$ be a continuous function, and let $\phi:(-\epsilon,\epsilon) \to \mathbb{R}$ be defined in some ...
3
votes
2answers
102 views

Evaluate $\int^0_1 \frac{\ln(t)}{1-t^2}dt$

Problem: Evaluate$$\int^0_1 \dfrac{\ln(t)}{1-t^2}dt$$ This actually came up while solving another Integral. It was suggested that I use a Binomial Series, but unfortunately, I couldn't ...
2
votes
3answers
67 views

Integral test for convergence of $\frac{1}{\ln x}$ [closed]

I want to know if $$\int_0^1 \frac{1}{\ln x}\, dx$$ converges or not.