Questions about the evaluation of specific definite integrals.

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3
votes
1answer
86 views

Evaluating $\int^b_a \frac{dx}{x}$ from the definition of the integral

I know that $$\int^b_a \frac{dx}{x}=\ln b-\ln a$$ I'm trying to evaluate this integral using the same method used in this answer: http://math.stackexchange.com/a/873507/42912 My attempt $\int^b_a ...
2
votes
0answers
70 views

Integral $\int_{0}^{\pi/2} \arctan \left(2\tan^2 x\right) \mathrm{d}x$

The following integral may seem easy to evaluate ... $$ \int_{0}^{\Large\frac{\pi}{2}} \arctan \left(2 \tan^2 x\right) \mathrm{d}x = \pi \arctan \left( \frac{1}{2} \right). $$ Could you prove ...
2
votes
2answers
75 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
1answer
27 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 ...
2
votes
3answers
24 views

Integration Proofs

Prove Show that for $x\geq 1$ $\displaystyle\left|\int_1^x \frac {\sin(t)}{t} dt\right|$ $ \leq \ln(x) $ Attempt: $\displaystyle\left|\int_1^x \frac {\sin(t)}{t} dt\right|$ $ $ $ \leq $ ...
1
vote
2answers
37 views

Proof that $\log(a^b) = b\log a$ when $\log$ is defined by an integral

When $\log a$ is defined as $\displaystyle\int_1^a\frac{dx}x$, then how does one prove that $\log(a^b)=b\log a$? I will post an answer here that is identical to an answer I posted to another ...
0
votes
2answers
30 views

Convergence of improper integral of $\ln f(x)$

Is there something know about the convergence of $\int_0^1 \ln f(x)dx $ for $f(x)$ continous on $\left(0,1\right)$ and both limits exists, i.e. $\lim_{x\to 0} f(x)$ and $\lim_{x\to 1} f(x)$ ? I ...
8
votes
2answers
156 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
108 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
2answers
29 views

Integral expression $P=x^3+x^2+ax+1$, …

I am trying to solve the following problem: Consider the integral expression $P=x^3+x^2+ax+1$, where $a$ is a rational number. At $a = ?$ the value of $P$ is a rational number for any x which ...
2
votes
4answers
114 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 ...
0
votes
1answer
24 views

Where is the error in this parameterization?

The problem is thus: find $$\int_c (x+2y)\mathrm{d}x+x^2\mathrm{d}y \space \mathrm{where \space C \space consists \space of \space \space line \space segments \space} (0,0)\space \mathrm{to} \space ...
4
votes
1answer
103 views

What is the error in this line integral?

C is the arc of the curve $y=\sqrt{x}$ from $(1,1)$ to $(4,2)$. Find $$\int_cx^2y^3-\sqrt{x}\space\mathrm{d}y$$Looks simple enough. I take $x=t$ and $y=\sqrt{t}$. This leaves $$\int_1^2[t^2\cdot ...
2
votes
1answer
54 views

Integral involving Gamma Function

I am solving the following integral: $$ \int_{-1}^{K} u^B e^{-u} du $$ The solution of the integral is a lower incomplete Gamma Function if -1 is replaced with 0. Can anybody help me in solving the ...
1
vote
2answers
102 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
176 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
105 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 ...
1
vote
2answers
65 views

Triple Integral exercise

Calculate $\int\int\int_Dz\;dxdydz$ if $D$ is the region inside $z=0,z=\sqrt{x^2+y^2}$ and $x^2+y^2=1$. I would like to know if the answer I got is right. This is what I did: $(1)$ Change to ...
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 ...
3
votes
0answers
47 views

Prove this polylogarithmic integral has the stated closed form value

Question. Prove the following polylogarithmic integral has the stated value: $$I:=\int_{0}^{1}\frac{\operatorname{Li}_2{(1-x)}\log^2{(1-x)}}{x}\mathrm{d}x=-11\zeta{(5)}+6\zeta{(3)}\zeta{(2)}.$$ ...
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
138 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
30 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
192 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 ...
0
votes
2answers
61 views

How to prove $ \int \sqrt{a^2-u^2}du $

How can I prove that the following definite integral? $$ \int \sqrt{a^2-u^2}du = \frac{1}{2}\left[u\sqrt{a^2 - u^2} + a^2 \arcsin\left(\frac{u}{a}\right)\right] +C$$
0
votes
0answers
12 views

Definite integral of product of regularized incomplete beta functions

I'm trying to find $$\int_{0}^{1} \! {I_x'(\alpha_1,\beta_1) I_x(\alpha_2,\beta_2) \dots I_x(\alpha_n,\beta_n)} \,\mathrm{d} x$$ where $\alpha_i, \beta_i$ are constant. Note the first term is ...
-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: ...
2
votes
1answer
35 views

Upper bound the integral or its PV (or prove that it diverges)

I need help in finding an upper bound the following integral (or its Cauchy Principal Value): $$ \int_0^1 \sqrt{x} \frac{|\ln(\frac{(y^{-1}-1)}{(x^{-1}-1)})|}{|x-y|} dx $$ This integral arises as ...
0
votes
1answer
20 views

Finding the boundaries of integration when calculating P(X + Y > a) or P(X + Y < b) (Jointly Distributed Continuous Random Variables)

I have a problem on setting the boundaries of integration when I'm trying to find probabilities like $P(X + Y > a)$ or $P(X + Y < b)$. For example, when I have $f(x,y) = \frac {x} {5}\ +\frac ...
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
189 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 ...
3
votes
2answers
46 views

Comparison of Newton-Cotes Quadrature and Gaussian Quadrature formulas

Newton-Cotes quadrature formulas are a generalization of trapezoidal and Simpson's rule. The trapezoidal rule involves $2$ points, Simpson's rule involves $3$, and in general Newton-Cotes formulas ...
0
votes
5answers
53 views

Proving the indefinite integral $\int \frac{1}{u(a+bu)}du $

How can I prove that the definite integral: $\int \frac{1}{u(a+bu)}du $ is equal to: $\frac{1}{a} ln|\frac{u}{a+bu}| +C $
0
votes
1answer
34 views

Definite integral and Riemann sum

$ \int_0^1 x^2 dx $ n is the number of pieces to cut into For $ (x_i - x_{i-1}) = \frac {0+1}{n} $ and $ x_i = \frac in $ Using the right Riemann sum: $ \Sigma_{i=1}^n f(x_i)*(x_i - x_{i-1}) $ $ ...
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. ...
-1
votes
1answer
42 views

Value of line integral [on hold]

Let $C=\{(x,y)\in \Bbb R^2:~\max\{|x|,|y|\}=1\}.$ The value of the line integral $$\oint_C(xy^2+2y+sin(e^x))dx+(x^2y+cos(e^y))dy$$ is?
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 ...
9
votes
0answers
143 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? $$\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$$