0
votes
1answer
40 views

Integral with quadratic square root inside trigonometric functions

Is there anyway to solve $\displaystyle \int t \frac{\sin \left(\frac{t}{2} \sqrt{ a \left(t+ \frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a}}\right) }{ \sqrt{ a \left(t+ ...
2
votes
2answers
20 views

Definite integral of trig function

I'm looking for some assistance on the following problem: Let $$ T(x) = \int_{4r^3}^{4} tsin(t^3)dt $$ Find $$T'(r)$$ I'm struggling to find the antiderivative of the sine function, particularly as ...
3
votes
2answers
53 views

Closed form of $\int_0^1 \frac{\operatorname{Li}_2\left( \sqrt{t} \right)}{2 \, \sqrt{t} \, \sqrt{1-t}} \,dt $

I'm looking for a closed form of this integral. $$I = \int_0^1 \frac{\operatorname{Li}_2\left( \sqrt{t} \right)}{2 \, \sqrt{t} \, \sqrt{1-t}} \,dt ,$$ where $\operatorname{Li}_2$ is the dilogarithm ...
0
votes
0answers
20 views

Integral analysis

So lately I had to work out mathematically the predicted relation between radiation intensity (I) and thickness of the obstacle (A). However, I found on the internet some notations using integrals ...
4
votes
0answers
44 views

Closed form for integral $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $

I'm looking for a closed form of this definite iterated integral. $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $$ From Vladimir ...
1
vote
0answers
22 views

Looking for advice with the following integral

I have the following integral to evaluate: $$ \frac{1}{f(t)}\int_0^t t^m (t + n)^o \sin(pt) \mathrm{d}t \quad m,n,o,p \in \mathbb{R}$$ I'm unable to proceed with this integral as it is non-trivial. ...
1
vote
1answer
58 views

Numerical value of $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $

Could somebody give me a numerical value for this integral? $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $$
0
votes
0answers
13 views

Calculus Single Variable: Find max and min of hard to graph function

Consider the function F defined by F(x)= integral from 0 to x of $t|sint(t)|dt$. Find the absolute maximum value and absolute minimum value of y=f(x). I know there's one at x= zero but the ones ...
10
votes
0answers
93 views
+200

Integral $\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$

I decided to follow a recent trend and ask a question about logarithmic integrals :) Is there a closed form for this integral? $$\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$$
1
vote
1answer
21 views

order of integrals with independent limits

I was wondering if the following is true assuming that the limits are independent (like constants) $$ \int_{\alpha}^{\beta} \int_{\gamma}^{\psi} {xy} dx dy = \int_{\gamma}^{\psi} ...
7
votes
1answer
44 views

Motivation behind parameters

This article shows a technique of evaluating a definite integral by introducing a suitable parameter. This however doesn't throw light on motivation for introducing that particular parameter. For ...
11
votes
2answers
131 views

Evaluating $\int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$

Good evening! I want to compute the integral $\displaystyle \int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$. However I find it extremely difficult. What I've tried is rewritting it as: ...
1
vote
0answers
49 views

Find $\int \tan(\tan x)\hspace{1mm}dx$

Find $\int \tan(\tan x)\hspace{1mm}dx$ This is an Interesting problem, which I have been trying from different directions, nothing seems to work, its been a day on this one. Can anyone figure out ...
0
votes
1answer
35 views

Finding area between two cosine curves

I must to find the area between these two curves: $$y = 2 \cos 7x, y = 2 − 2 \cos 7x$$ $$0 ≤ x ≤ π/7$$ And this is all I have so far: $$ 2\cos7x=2-2\cos7x $$ $$4\cos7x=2$$ $$\cos7x=1/2$$
15
votes
1answer
200 views
+300

Closed form for ${\large\int}_0^1\frac{\ln^{\color{magenta}3}x}{\sqrt{x^2-x+1}}dx$

This is a follow-up to my earlier question Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$. Is there a closed form for this integral? ...
2
votes
1answer
45 views

How to simplify this complex integral? [on hold]

How to approximate this integral as a function of a and b? $$\int_0^\pi\int_0^{2\pi}\sqrt{(a-b\sin\varphi\cos\theta)^2+(b\cos\varphi)^2+(b\sin\varphi\sin\theta)^2}d\theta d\varphi$$ where a and b ...
3
votes
4answers
225 views

Finding the definite integral of a function that contains an absolute value

The integral in question is this: $\int_{-2\pi}^{2\pi}xe^{-|x|}$ My attempt: Since there is a modulus, we split it up into cases. I'm not really sure which cases to split it into, do I just ...
0
votes
0answers
31 views

Could someone help find the shell height?

I am trying to solve this problem and have been going at it for 3 hours and not getting anywhere. I think I am suppose to have everything in terms of y but the x equals functions are throwing me off. ...
-1
votes
2answers
33 views

Find $\int t\sin^{-1}t\hspace{1mm}dt$

Find $\int t\sin^{-1}t\hspace{1mm}dt$ How do we approach this question, is there a simple way to integrate
1
vote
1answer
38 views

Integral from 0 to 16 of $\sqrt{x}/(x-4)$?

$$\int_{0}^{16}\frac{\sqrt{x}}{x-4}dx$$ So I'm letting $u=\sqrt{x}$, $du=1/2\sqrt{x}$, $u^2=x$ and $dx=2\sqrt{x}du$. I just don't really know what to do from here. I am trying different things and ...
1
vote
4answers
84 views

Does the following integral converge: $\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$

Does the following integral converge: $$\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$$ I suppose we have to solve such problems by comparison test. All the integrals I tried so far do not fit the ...
0
votes
0answers
58 views

Integral $\int_0^1 \frac{\sqrt{1-x}}{\sqrt{1+x^2}} dx$

Looking for a closed-form of this integral. $$I=\int_0^1 \frac{\sqrt{1-x}}{\sqrt{1+x^2}} dx$$ I'm looking for a closed-form of $I$ without using Meijer G-function, elliptic integrals or generalized ...
1
vote
2answers
52 views

Find $\int_0^{\pi}\sin^2x\cos^4x\hspace{1mm}dx$

Find $\int_0^{\pi}\sin^2x\cos^4x\hspace{1mm}dx$ $ $ This appears to be an easy problem, but it is consuming a lot of time, I am wondering if an easy way is possible. WHAT I DID : Wrote this as ...
19
votes
2answers
330 views
+300

Closed-forms for several tough integrals

These integrals came up in the process of finding solution to Vladimir Reshetnikov's problem. I wonder if there are closed-forms for the following integrals: \begin{array}{1,1} &[\text{1}] ...
5
votes
2answers
55 views

Integral with rational functions of powers and exponentials

Any ideas how to solve: \begin{equation} \int_0^\infty x^{n+\frac{1}{2}} (e^{a x }-1)^{-\frac{1}{2}} e^{i x t} dx \end{equation} where $a$ and $t$ are real, positive constants; $n$ is a positive ...
0
votes
4answers
154 views

Find $\int_0^{1/2} \sqrt{1+\sqrt{1-x^2}}\hspace{1mm}dx$ [on hold]

Find $\int_0^{1/2} \sqrt{1+\sqrt{1-x^2}}\hspace{1mm}dx$ How do we approach this problem, can someone explain
16
votes
3answers
252 views

Integral $\int_{0}^1\frac{\ln\frac{3+x}{3-x}}{\sqrt{x(1-x)}}dx$

I have a problem with the following integral: $$ \int_{0}^{1}\ln\left(\,3 + x \over 3 - x\,\right)\, {{\rm d}x \over \,\sqrt{\,x\left(\,1 - x\,\right)\,}\,} $$ The first idea was to use the ...
21
votes
3answers
378 views

Integral of Combination Log and Inverse Trig Function

Does the following integral have a closed-form ?: \begin{equation} \int_{0}^{1}{\ln\left(\,x\,\right) \over 1 + x}\,\arccos\left(\,x\,\right) \,{\rm d}x \end{equation} This integral has been ...
6
votes
4answers
124 views

Calculus Question: Improper integral $\int_{0}^{\infty}\frac{\cos(2x+1)}{\sqrt[3]{x}}dx$

How to evaluate integral $$\int_{0}^{\infty}\frac{\cos(2x+1)}{\sqrt[3]{x}}dx?$$ I tried substitution $x=u^3$ and I got $3\displaystyle\int_{0}^{\infty}u \cos(2u^3+1)du$. After that I tried to use ...
2
votes
0answers
43 views

How do I solve this tricky definite integral ?! [duplicate]

$$I=\int\limits_0^1\dfrac{x^2-1}{\ln x}\mathrm dx$$ I tried numerous substitutions but nothing seems to work.. any ideas ???!
7
votes
0answers
169 views

How to evaluate the integral $e^{-(c\ln(\frac{1}{x}))^s} dx$?

Can anyone help me evaluate $$\int_{\alpha}^1 \exp{\left\{-\left(c\ln\left(\frac{1}{x}\right)\right)^s\right\}} dx$$, Where $0 \leq \alpha \leq 1$ and $s \in \mathbb{R}$. I tried changing ...
1
vote
2answers
72 views

Evaluating $\;\int_{1}^{\ln3}\frac{e^x - e^{2x}}{(1 + e^x)}\,dx$

Find $\int_{1}^{\ln3}(e^x - e^{2x})/(1 + e^x)dx$. I looked through my notes for integration techniques and thought I could try a $u$ substitution but whatever I set $u$ to I can't seem to ...
1
vote
3answers
40 views

Compute variance, using explicit PDF

I'm trying to get $\text{Var}(x)$ of $f(x) = 2(1+x)^{-3},\ x>0$. Please tell me if my working is correct and/or whether there is a better method I can use to get this more easily. $$ ...
0
votes
2answers
33 views

Help finding k. Issue with integration

Let the continuous random variable $X$ have a probability density function $f(x)$ such that $$f(x) = k(1+x)^{-3}, x>0$$ $=0$ elsewhere Find k This is what I tried: $\int_0^\infty k(1+x)^{-3}dx ...
0
votes
1answer
75 views

How to evaluate a definite integral involving the product of two sines?

The input signal for a given electronic circuit is a function of time $V_{in}(t)$. The output signal is given by $$V_{out}(t) = \int_0^t \sin(t-s) V_{in}(s)\,ds$$ Find $V_{out}(t)$ if ...
1
vote
1answer
50 views

How to integrate $\sqrt{1+(2/3)x}$?

How would you solve the following (step by step please!): $$\int^6_5\sqrt{1+\frac23x}\ dx$$ I started with $u=1+\frac23x$, $du=\frac23\,dx$, now what?
2
votes
3answers
45 views

Improper integral of rational function $k^2/(1+a^2k^2)^2$

I've got the integral $\int^\infty_{-\infty} dk \frac{k^2}{(1+a^2 k^2)^2}$ where $a$ is a real number. I can't seem to find a $u$-substitution or trigonometric substitution that will work. Any ...
0
votes
2answers
171 views

Volume of Solid Revolution

region bounded by $$y=x$$ and $$y=x^2$$ a) find the volume of the solid of revolution formed by revolving R region about the line $x=2$. b) find the volume of the solid of revolution formed by ...
0
votes
2answers
53 views

If a function $f$ is bounded in $[a,b]$ and if for each $x\in(a,b)$, $f$ is integrable over $[x,b]$, Then $f$ is integrable over $[a,b]$

I got this question which popped up in my mind: Prove or disprove: If a function $f$ is bounded in a closed interval $[a,b]$ and if for each $x\in(a,b)$, $f$ is integrable over $[x,b]$, Then $f$ is ...
30
votes
2answers
689 views

Closed form for ${\large\int}_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx$

Here is another integral I'm trying to evaluate: $$I=\int_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx.\tag1$$ A numeric approximation is: ...
0
votes
0answers
28 views

CAS Can't Solve the Integral

I am trying to solve this integral; $$ \int^a_{-a}\frac{E[\frac{-(4b_1 b_2)}{((x+z)^2+(b_1-b_2)^2 )}]}{((x+z)^2+(b_1+b_2)^2)\sqrt{((x+z)^2+(b_1-b_2)^2 )}}dx $$ E is the complete elliptic integral of ...
4
votes
1answer
67 views

Evaluating the integral $\int_{-1}^{1}\frac{x^2\,dx}{e^x+1}$

Can this integral be evaluated properly by using highschool integration skills ? $$\int_{-1}^{1}\dfrac{x^2}{e^x+1}\,dx$$ (Original image at http://i.stack.imgur.com/AqOsX.png) Judging from what I ...
5
votes
2answers
309 views

A logarithmic integral

How can I evaluate following logarithmic integral: $$\int\limits_0^1 \frac{\ln x\ln ( 1 - zx )}{1 - x} dx$$
21
votes
4answers
540 views

Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$

I want to find a closed form for this integral: $$I=\int_0^1\frac{\ln^2x}{\sqrt{x^2-x+1}}dx\tag1$$ Mathematica and Maple cannot evaluate it directly, and I was not able to find it in tables. A numeric ...
1
vote
2answers
60 views

Question about the limit definitions of derivative and definite integral

Actually this question may look simple and basic, but it is about something which bugs me for a long time, since when I took my first calculus classes. The limit $\lim_{x \to a}f(x) = L$ is defined ...
10
votes
8answers
239 views

Evaluate $ \int_{0}^{1} \ln(x)\ln(1-x)\,dx $

Evaluate the integral, $$ \int_{0}^{1} \ln(x)\ln(1-x)\,dx$$ I solved this problem, by writing power series and then calculating the series and found the answer to be $ 2 -\zeta(2) $, but I don't ...
11
votes
4answers
476 views

Looking for closed-forms of $\int_0^{\pi/4}\ln^2(\sin x)\,dx$ and $\int_0^{\pi/4}\ln^2(\cos x)\,dx$

A few days ago, I posted the following problems Prove that \begin{equation} \int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}\\[20pt] -\int_0^{\pi/2}\ln^3(\cos ...
1
vote
1answer
52 views

Why is $\int_0^{2\pi} e^{i\,k\rho[\sin\alpha\cos\alpha-\sin\theta\cos(\phi-\beta)]}\mathrm{d}\beta = 2\pi J_0(k\rho\xi)$?

The following is an integral in Jackson Classical Electrodynamics (3rd ed.). In equation (10.112) the integral $$ \int_0^{2\pi} ...
4
votes
4answers
168 views

Prove that $\int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}$

Prove that \begin{equation} \int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24} \end{equation} I tried to use by parts method and ended with \begin{equation} \int \ln^2(\cos ...
17
votes
2answers
279 views

Prove $\displaystyle \int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2 $

How to prove\begin{equation} \int_{0}^{\pi/2} \ln \left(x^{2} + (\ln(\cos x))^2 \right) \, dx=\pi\ln\ln2 \end{equation} I don't know how to answer it. When I asked this integral to my brother, ...