2
votes
2answers
71 views

A closed form of $\int_0^1\frac{\ln\ln\left(\frac{1}{x}\right)}{x^2-x+1}dx$

This integral has been bugging me since yesterday: $$\int_0^1\frac{\ln\ln\left(\frac{1}{x}\right)}{x^2-x+1}dx$$ I've tried substitution $y=\frac{1}{x}$ and $e^y=\frac{1}{x}$, but those didn't ...
0
votes
0answers
20 views

Continuity of $K(x,y)$ satisfying $g(x)= \int_0^1 \! K(x,y) f(y)\ \mathrm{d}x $ and $ \frac{d^3g}{dx^3} = f$

$g(x)$ is defined as the following : $$g(x)= \int_0^1 \! K(x,y) f(y)\ \mathrm{d}x $$ where $K(x,y)$ is continuous in $ 0 \leq x \leq 1 $ , $ 0 \leq y \leq 1 $, and $f(x)$ is continuous in $ 0 \leq x ...
8
votes
2answers
152 views

A Binet-like integral $\int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{x^s }{1-x}\mathrm{d}x$

I met this integral $$ \int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{ \mathrm{d}x}{1-x} \qquad (*) $$ while evaluating this log-cosine integral. I made several ...
5
votes
1answer
105 views

Evaluation of $\int_{0}^{\frac{\pi}{2}}\frac{\sin (2015x)}{\sin x+\cos x}dx$

Evaluation of $\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\sin (2015x)}{\sin x+\cos x}dx$ $\bf{My\; Try:}$ Let $\displaystyle I = \int_{0}^{\frac{\pi}{2}}\frac{\sin (2015x)}{\sin x+\cos x}dx\tag{1}$ ...
3
votes
3answers
91 views

$\int_{0}^{\pi/2}\ln\left(1+4\sin^4 x\right)\mathrm{d}x$ and the golden ratio

We already know that, for any real number $t$ such that $t\geq-1$, $$ \int_{0}^{\pi/2} \ln \left(1+t \sin^2 x\right) \mathrm{d}x = \pi \ln \left( \frac{1+\sqrt{1+t}}{2} \right). $$ Prove that ...
2
votes
4answers
88 views

Using integral definition to solve this integral

I'm trying to solve this question using the definition of integral: $$\int^5_2 (4-2x)dx$$ Definition of integral: We define first the inferior and superior sum: Let $f:[a,b]\to \mathbb R$ be a ...
1
vote
2answers
100 views

Evaluate $\int_0^1 \sqrt{2x-1} - \sqrt{x}$ $dx$

I'm trying to calculate the area between the curves $y = \sqrt{x}$ and $y= \sqrt{2x-1}$ Here's the graph: I've already tried calculating the area with respect to $y$, i.e. $\int_0^1 ...
7
votes
2answers
171 views

Proving that $\int_0^1\frac{x \log^2(1-x)}{1+x^2} \ dx = \frac{35}{32}\zeta(3)+\frac{1}{24}\log^3(2) -\frac{5}{96} \pi^2 \log(2)$

Could we possibly prove this result without using the polylogarithm? I know how to do it by polylogarithm means, but I want a different way. Is that possible? $$\int_0^1\frac{x \log^2(1-x)}{1+x^2} ...
2
votes
4answers
104 views

How to prove $\int_0^\pi \frac{dx}{2+2\sin x+\cos x}=\log3$?

How can we prove that: $$\int_0^\pi \frac{dx}{2+2\sin x+\cos x}=\log3$$ I don't have any ideas, the $f(\pi-x)$ thing doesn't work as well. Please help :)
2
votes
2answers
47 views

Examples of interesting integrable functions with at least 2 fixed points and an explicit inverse

What are some interesting functions I can use to demonstrate this integration trick: $$\int_a^b [f(x)+f^{-1}(x)]=b^2-a^2$$ I would like to know of some interesting functions where this trick is not ...
6
votes
1answer
63 views

Integral involving square root of sine and cosine

Is there any closed formula for $$ \int_{0}^{\pi/2} \dfrac{e^{-x}\sqrt{\cos x}\ dx}{\sqrt{\cos x} + \sqrt{\sin x}} $$ I know $$ \int_{0}^{\pi/2} \dfrac{\sqrt{\cos x}\ dx}{\sqrt{\cos x} + \sqrt{\sin ...
11
votes
3answers
217 views

Suggestion for Computing an Integral

Let $$A=\left\{(x,y,z)\in \mathbb R^3:\dfrac{x^2}{2}+\dfrac{y^4}{4}+\dfrac{z^6}{6}\leq1\right\}.$$ Then I want to compute the following integral: ...
12
votes
1answer
135 views

$\int_0^{2\pi}e^{\cos x}\cos(\sin x)dx$ [duplicate]

$$\int_0^{2\pi}e^{\cos x}\cos(\sin x)dx$$ I tried Integration by parts but failed. Wolfram alpha gives answer in decimal points which are same as of $2\pi$. Any hints or suggestions will be helpful.
2
votes
1answer
29 views

Check the properties of the eigenfunction corresponding to the distinct eigenvalues of an integral equation

Let $\lambda_1, \lambda_2$ be eigenvalues and $f_1 , f_2$ be the corresponding eigenfunctions for the homogeneous integral equation \begin{align} \phi(x) - \lambda \int_0^1 (xt +2x^2) \phi(t) ...
3
votes
5answers
191 views

Calculating the area

For the two graphs $ \frac{x^3+2x^2-8x+6}{x+4} $ and $ \frac{x^3+x^2-10x+9}{x+4} $, calculate the area which is confined by them; Attempt to solve: Limits of the integral are $1$ and $-3$, so I took ...
2
votes
1answer
86 views

Improper integral $\int_{0}^{\pi} \frac{x}{\sin x} dx$

Find out whether or not the following integral exists $$\int_{0}^{\pi} \frac{x}{\sin x} dx.$$ I'm pretty sure this integral doesn't exist but I can't seem to find a good way to prove this. It ...
0
votes
0answers
35 views

The negative integral meaning

Whenever I take a definite integral in aim to calculate the area bound between two functions, what is the meaning of a negative result? Does it simly mean that the said area is under the the x - axis, ...
3
votes
1answer
30 views

Volume of a solid(between two planes)?

A solid lies between planes perpendicular to the y-axis at $ y=0$ and $y=1$. The cross-sections perpendicular to the y-axis are circular disks with diameters running from the y-axis to the parabola ...
-1
votes
1answer
59 views

How to find the volume of a solid [on hold]

Suppose the solid $E$ is given by $$E=\{(x,y,z): (x^2+y^2+z^2+8)^2 \leq 36(x^2+y^2)\}.$$ Find the exact volume of $E$.
0
votes
1answer
70 views

Show that $\int_{\pi/4}^{\pi/2} \frac{\sin x}{x}\,dx\leq \frac{\sqrt{2}}{2}$

Show that $$\int_{\pi/4}^{\pi/2} \dfrac{\sin x}{x}\,dx\leq \dfrac{\sqrt{2}}{2}$$ Any Ideas, how to start ?!
2
votes
4answers
129 views

What is the integral of x/ln(x)?

Well, I'm french so excuse me if I make some mistakes in english... I have to calculate this integral : $$ \int_{e}^{2e} \frac{x}{\ln(x)} dx $$ But I don't know how, can you help me please? Thank ...
5
votes
1answer
47 views

Maximum value problem

A function $\hspace{0.1cm}$$f:[0,1]\to[-1,1]$$\hspace{0.1cm}$ satisfying$\hspace{0.1cm}$ $|f(x)|\leq x$$\hspace{0.1cm}$ $\forall x\in[0,1]$. Then find the maximum value of: ...
6
votes
3answers
193 views

Integral $\int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$

Calculate the following integral: \begin{equation} \int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx \end{equation} I am having trouble to calculate the integral. I ...
-1
votes
2answers
62 views

Prove that $\displaystyle\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$

Prove that $\displaystyle\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$ Note that $c$ need not belong to $(a, b)$ And $f(x)$ is a continuous function. All ideas are appreciated.
7
votes
2answers
188 views

Another integral for $\pi$

Here is a new integral for $\pi$. $$\int_{0}^{1}\sqrt{\frac{\left\{1/x\right\}}{1-\left\{1/x\right\}}}\, \frac{\mathrm{d}x}{1-x} = \pi $$ where $\left\{x\right\}$ denotes the fractional part of ...
1
vote
1answer
71 views
5
votes
3answers
153 views

Examples of “difficult” integrals with are easier to solve with a series?

Yesterday someone posted an extremely elegant solution to a seemingly bizarre series where the integral: $$\int_{0}^{1} x^{m}\ dx = \frac{1}{m + 1}$$ was utilized. Oftentimes one will also ...
1
vote
1answer
48 views

A definite integral involving $\exp(-a\cosh x)$

Wolfram alpha can not integrate the following integral: $$\int_0^\infty x^n \exp(-a\cosh x)dx$$ Thanks- Mike
2
votes
1answer
19 views

differential in integration cancelled but variable endpoint is changed

In kinetic energy equation in wiki. I have difficulty problem how endpoint in integral change from t to v. This doesn't look like it is using substitution method. ...
0
votes
1answer
26 views

How to compute the area of this set in the plane?

Let $f$ be a non-negative function which is defined, bounded, and integrable on a closed interval $[a,b]$, and let $$ S \colon= \{\ (x,y) \ | \ a \leq x \leq b, \ 0 \leq y < f(x) \ \}. $$ Then is ...
8
votes
0answers
142 views

The closed form of $\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$

What tools, ways would you propose for getting the closed form of this integral? $$\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$$
2
votes
6answers
136 views

What is an easy way to integrate $\int_0^5 \frac{v^3 }{2\sqrt{25-v^2}} dv$?

This does not appear to be a difficult integral. I am wondering if there was an easy way to do it.
2
votes
4answers
506 views

Integration by substitution, why do we change the limits?

I've highlighted the part I don't understand in red. Why do we change the limits of integration here? What difference does it make? Source of Quotation: Calculus: Early Transcendentals, 7th Edition, ...
2
votes
1answer
119 views

Use discrete proof to show that $\int f^2 \int g^2 \geq (\int fg)^2$

One proof of the Schwarz inequality on $\mathbb{R}^n$ is to note that $$(\sum x_i^2)(\sum y_i^2) = (\sum x_i y_i)^2 + \sum_{i<j}(x_iy_j - x_jy_i)^2.$$ Spivak's Calculus, 4th ed., exercise ...
4
votes
2answers
639 views

The area of circle

The question is to prove that area of a circle with radius $r$ is $\pi r^2$ using integral. I tried to write $$A=\int\limits_{-r}^{r}2\sqrt{r^2-x^2}\ dx$$ but I don't know what to do next.
3
votes
3answers
61 views

How do i prove $\int_{0}^{2a}f(x) dx = \int_{0}^{a}f(x)dx +\int_{0}^{a}f(2a-x)dx$

I am stuck on this one, I was able to prove $\int_{0}^{a}f(x) dx = \int_{0}^{a}f(a-x)dx $ just by using substitution with $t = a - x$, but I'm not sure about this one, I've tried quite a few ...
9
votes
3answers
146 views

Prove $\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$

I have in trouble for evaluating following integral $$\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$$ It seems really easy, but I ...
2
votes
3answers
147 views

Trouble computing this double integral

$$\iint_R xe^{xy}~\mathrm{d}A \qquad 0\le x\le 2 \quad 0 \le y \le 1$$ Today I started learning about double integrals on a class I am taking, had good understanding on single-variable integrals but ...
10
votes
2answers
261 views
+50

A closed form for $\int_{0}^{\pi/2} x^3 \ln^3(2 \cos x)\:\mathrm{d}x$

We already know that \begin{align} \displaystyle & \int_{0}^{\pi/2} x \ln(2 \cos x)\:\mathrm{d}x = -\frac{7}{16} \zeta(3), \\\\ & \int_{0}^{\pi/2} x^2 \ln^2(2 \cos x)\:\mathrm{d}x = ...
6
votes
1answer
83 views

Evaluate: $I = \int^{\pi/2}_0 (\sqrt{\sin x}+\sqrt{\cos x})^{-4}dx$

Evaluate : $$I = \int_{0}^{\Large\frac\pi2} (\sqrt{\sin x}+\sqrt{\cos x})^{-4}\ dx$$ Attempt : \begin{align} I&=\int_{0}^{\Large\frac\pi2} (\sqrt{\sin x}+\sqrt{\cos x})^{-4}\ dx\\ ...
3
votes
2answers
115 views

Find $\lambda$ if $\int^{\infty}_0 \frac{\log(1+x^2)}{(1+x^2)}dx = \lambda \int^1_0 \frac{\log(1+x)}{(1+x^2)}dx$

Problem : If $\displaystyle\int^\infty_0 \frac{\log(1+x^2)}{(1+x^2)}\,dx = \lambda \int^1_0 \frac{\log(1+x)}{(1+x^2)}\,dx$ then find the value of $\lambda$. I am not getting any clue how to proceed ...
1
vote
0answers
54 views

Analytical evaluation of integral

I would like to evaluate the following integral analytically, but Mathematica does not give me an answer: $$ \int_0^1 dr \ e^{(1-2r)x^2} \left[p(r,x) Y_0\left(2x^2\sqrt{r-r^2}\right)+q(r,x) ...
0
votes
1answer
50 views

How do they integrate this exponential?

Below, I tired to integrate te^(-j2pi*t) from 0 to 1. But am not getting what my professor got for n not equal to zero, which is also shown. I tried LIATE but am always getting something with an ...
1
vote
0answers
29 views

How to establish the equivalence of these two statements about integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a, b]$. Then we have $$ \int_{ka}^{kb} s\left(\frac{x}{k}\right) \ dx = k \int_a^b s(x) \ dx $$ for every $k ...
0
votes
1answer
31 views

How to establish this equivalence for integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a,b]$. Then we have $$\int_{a}^{b} s(x) \ dx = \int_{a+c}^{b+c} s(x-c) \ dx.$$ Second Statement: Let $s$ be ...
7
votes
2answers
162 views

Integral: $\int_0^{\infty} \cos\left(\frac{a^2}{x^2}-b^2x^2\right)\,dx$ for $a,b>0$

I tried this: $$\int_0^{\infty} \cos\left(\frac{a^2}{x^2}-b^2x^2\right)\,dx=\Re\left(\int_0^{\infty} e^{-ib^2x^2+ia^2/x^2}\,dx\right)=\Re\left(\int_0^{\infty} ...
1
vote
1answer
66 views

How to calculate an integral

I wonder how the integral $$\int_{-1}^1 \! \int_0^{\sqrt{1-x^2}} \! \int_0^{\sqrt{1-y^2-x^2}} \! 1 \, dz \, dy \, dx $$ Any ideas?
-2
votes
1answer
62 views

Integral question challenge [duplicate]

I try to find a reasonable solution for this equation but i couldent I try to study lots of material but i couldent solve it. I am a high school student and try to learn. Integral cos(log x)dx
-1
votes
3answers
110 views

Integral big question

Anyone could help me to solve this equation I try to study lots of material but I coulden't solve it. I am a high school student and try to learn. $\displaystyle\int \cos(\ln(x))dx$?
4
votes
6answers
104 views

If $\lim\limits_{x \to \infty} f(x) = 1$, can we have function $f(x)$, such that $\int_0^{\infty}f(x)dx$ converges

I know the Initiative answer, can anyone give a neat answer based on solid reasoning EDIT : $f(x)$ is continuous