Questions about the evaluation of specific definite integrals.

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2
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3answers
22 views

Integral of $\cos(\cos x)$ over $[0,2\pi]$

How to compute the following integral? $$\mathcal{J}_2=\int_{0}^{2\pi}\cos(\cos t)\,dt$$ I'm trying to compute this integral, but I have no idea of how to do it, can someone help me?
6
votes
2answers
51 views

Trigonometric functions expressed as definite integrals with Bessel functions

Prove that $$\frac{\sin(x)}{x}=\int_0^\frac{\pi}{2}J_0(x\cos(\theta))\cos(\theta)\,d\theta \tag{a}$$ $$\frac{1-\cos(x)}{x}=\int_0^\frac{\pi}{2}J_1(x\cos(\theta))\,d\theta \tag{b}$$ Hint: ...
4
votes
1answer
66 views

A closed-form of $\frac{1}{2}\int_0^\infty\left[\frac{x^2\cos x}{\cosh 2x-\cos x}-\frac{2x^2}{e^{4x}-2e^{2x}\cos x+1}\right]\,dx$

I am looking for a closed-form of this integral \begin{equation} \frac{1}{2}\int_0^\infty\left[\frac{x^2\cos x}{\cosh 2x-\cos x}-\frac{2x^2}{e^{4x}-2e^{2x}\cos x+1}\right]\,dx \end{equation} I ...
4
votes
0answers
66 views

is there closed form for $\int_0^{\pi/4}\exp(-\sum_{n=1}^{\infty}\frac{\tan^{2n}x}{n+a})dx$

Is there closed form for $$I(a)=\int_0^{\pi/4}\exp(-\sum_{n=1}^{\infty}\frac{\tan^{2n}x}{n+a})dx $$where is $a\in (-1,3)$ I've tried with $\tan x=u$ and I got the result of sum in term of ...
12
votes
2answers
179 views

Integral $\int_0^{\Large\frac{\pi}{4}}\left(\frac{1}{\log(\tan(x))}+\frac{1}{1-\tan(x)}\right)dx$

I am wondering if anyone would know how to evaluate this integral: $$\int_{0}^{\Large\frac{\pi}{4}}\left(\frac{1}{\log(\tan(x))}+\frac{1}{1-\tan(x)}\right)dx.$$ I've tried, unsuccessfully, the change ...
4
votes
3answers
69 views

Integration of $\int_0^{2\pi}(e^{\cos x}\cos x\sin x) \,dx$

Can anyone please help me with this integration: $$\int_0^{2\pi}(e^{\cos x}\cos x\sin x)\,dx$$ I am getting the answer as $0$ by using simply the properties of definite integral BUT the answer as ...
13
votes
2answers
86 views

Integrals of integer powers of dilogarithm function

I'm interested in evaluating integrals of positive integer powers of the dilogarithm function. I'd like to see the general case tackled if possible, or barring that then as many particular cases as ...
2
votes
2answers
90 views

Why must a continuous function be null if its definite integral is null? [duplicate]

Let $ f(x) = \begin{cases} f:[a,b] \rightarrow\mathbb R \\ \int_{a}^{b}f = 0 \end{cases}$. Prove: if $f$ is continuous, then $f\equiv 0$. I'm still trying to get the intuition on the situation. For ...
0
votes
2answers
9 views

Verify (Riemann) integrability proposition with counterexamples

Let $f[a,b]: \mathbb R \rightarrow\mathbb R$ be a (Riemann) integrable function such that $f \geq0$ and $\int_{a}^{b}f = 0$. Verify with counterexamples that the sentence aforementioned conditions do ...
0
votes
0answers
7 views

Simplifying general formula for fractional derivative by removed derivative of integral.

On the wikipage about fractional calculus, there's a general formula for the fractional derivative: $D^\alpha$ is the derivative operator. $$D^\alpha ...
6
votes
2answers
105 views

Integral $ \int_{0}^1 \sqrt{\frac{\ln{x}}{x^2-1}} dx$

Please help evaluating this integral $$ \large\int_{0}^1 \sqrt{\frac{\ln{x}}{x^2-1}} dx$$ Mathematica could not evaluate it in a closed form. Numerically it is about ...
3
votes
1answer
56 views

Nonsensical result in the midst of calculating an integral via substitution.

I was just calculating an integral via a trigonometric substitution and ended up with $\color{red}{ \text{something pretty nonsensical} }$ but $\color{blue}{ \text{reversing the substitution} }$ ...
2
votes
0answers
64 views

Contour Integral $ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$

I need help evaluating this with contour integration$$ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$$ I am not sure as to how to work with the branch cuts of both $\ln{x}$ and $\sqrt{1-x^2}$ ...
0
votes
0answers
23 views

6.17 Theorem : Show that $f \ \ \in \mathfrak R(\alpha)$ if and only if $ f\alpha' \ \ \in \mathfrak R$ ( walter rudin)

Question : Assume $\alpha$ increases monotonically and $\alpha' \in \mathfrak R$ on $[a,b]$. Let $f$ be a bounded real valued function on $[a,b]$. Then $f \in \mathfrak R(\alpha)$ if and only ...
0
votes
2answers
58 views

A identity relating a infinite series and a definite integral [duplicate]

Prove that, $$ \sum_{n=1}^{\infty} \frac{1}{n^n} = \int_{0}^{1} x^{-x}dx$$ I made no significant progress, I'm looking for hint/ideas to approach this problem. Thanks!
-2
votes
0answers
22 views

Riemann's sum inequality problem [on hold]

Iam having touble with a certain question on my assignment. I dont know how to replicate the math symbols on this site so I have jst put down a link to the full assignment: ...
7
votes
3answers
151 views

Hard Definite integral involving the Zeta function

Prove that: $$\displaystyle \int_{0}^{1}\frac{1-x}{1-x^{6}}{\ln^4{x}} \ {dx} = \frac{16{{\pi}^{5}}}{243\sqrt[]{{3}}}+\frac{605\zeta(5)}{54} $$ I was able to simplify it a bit by substituting ${y = ...
3
votes
5answers
138 views

Evaluate$ \int_0^{\frac{\pi}{2}} \ln(1+\cos x) dx$

Find the value of the integral $ \int_0^{\frac{\pi}{2}} \ln(1+\cos x) $ I tried putting $1+ \cos x = 2 \cos^2 \frac{x}{2} $, but am unable to proceed further. I think the following integral can be ...
3
votes
1answer
98 views

When may we ignore the limits of integration?

When we try to evaluate an integral such as, say $$\int_a^b{f(x)dx}$$ there is often the case that we can analytically find $$\int{f(x)dx}$$ a little faster (imagine leaving away the evaluation ...
0
votes
1answer
21 views

Antiderivative of unbounded function?

One way to visualize an antiderivative is that the area under the derivative is added to the initial value of the antiderivative to get the final value of the antiderivative over an interval. The ...
0
votes
0answers
42 views

Arc length for a function $f:\mathbb{R}^2 \to \mathbb{R}^2.$

Assume $f:\mathbb{R}^2 \to \mathbb{R}^2$ is $C^1.$ Is there a formula for the length of the subset of $\mathbb{R}^2$ given by $$ \{f(x,y) \in \mathbb{R}^2:a_1\leq x\leq b_1, a_2\leq y \leq b_2\} ? $$ ...
8
votes
2answers
178 views

Integral inequality: $\def\intd{\,\mathrm d}\int_a^b(f'(x))^2\intd x-2\big(f(a)+f(b)\big)^2\geq\frac8{(b-a)^2}\int_a^b(f(x))^2\intd x$

I have a problem which I think is wrong. Let $f: [a,b] \to \mathbb{R}$ be a differentiable function with $f'$ continuous such that $$\int_a^b f(x) \intd x = f\left(\frac{a+b}{2}\right) = 0$$ ...
4
votes
1answer
56 views

Can you prove a definite integral has no closed form?

It is a well known fact that some functions posses no closed form antiderivative yet still they have definite integrals that have a closed form. A classic example is the Gaussian integral ...
33
votes
5answers
785 views
+200

How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$

Please help me to find a closed form for this integral: $$I=\int_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx\tag1$$ I suspect it might exist because there are similar integrals having closed forms: ...
3
votes
2answers
86 views

cosine integral

Show that $$\int_0^x \frac{1-\cos(t)}{t}=\gamma+\ln(x)-\operatorname{Ci}(x)$$ where $$\operatorname{Ci}(x)=-\int_x^\infty \frac{\cos(t)}{t} \, dt$$ and gamma is an euler-mascheroni constant. I did as ...
2
votes
2answers
95 views

Is it true that $\int_0^1 \lfloor x^{-1} \rfloor^{-1} x^n dx = \frac{1}{n+1}(\zeta(2)+\zeta(3) + \dots + \zeta(n+2) ) - 1$?

This question is inspired by the formula $$\displaystyle\int_0^1{\left\lfloor{1\over x}\right\rfloor}^{-1}\!\!dx={1\over2^2}+{1\over3^2}+{1\over4^2}+\cdots = \zeta(2)-1,$$ see for instance this ...
0
votes
1answer
20 views

Derivative of an integral with variable in upper bound and a term of the integrand

So I want to take the first and second derivatives of a function g(Z) which is made up of several terms, one of which is where Z and H are our variables. Taking the derivative of this, it seems ...
1
vote
1answer
72 views

Prove that $\int_{-1}^1P_n^2(x)dx=\frac{2}{2n+1}$, where $P_n(x)$ is a Legendre polynomial.

Using Rodrigues' formula and integrating by parts $n$ times, prove that $$\int_{-1}^1P_n^2(x)dx=\frac{2}{2n+1}$$ where $P_n(x)$ is a Legendre polynomial. I tried this way Let $$f(x)=(x^2-1)^s$$ ...
3
votes
2answers
88 views

Prove: $\int_{0}^{1}\frac{\ln{x}\,\mathrm{d}x}{\sqrt[3]{x(1-x^2)^2}}\stackrel{?}{=}-\frac18\left[\Gamma{\left(\frac13\right)}\right]^3$

I'd like to evaluate the following definite integral: $$\int_{0}^{1}\frac{\ln{x}\,\mathrm{d}x}{\sqrt[3]{x(1-x^2)^2}}\stackrel{?}{=}-\frac18\left[\Gamma{\left(\frac13\right)}\right]^3.$$ ...
2
votes
0answers
97 views

Evaluating $\int_0^x \lvert \cos t \rvert dt$

in my mathbook there is given a solution to $$\int_0^x \lvert \cos t \rvert \, dt $$ but without any hints or tips. $$\int_0^x \lvert \cos t \rvert \, dt = \sin\left(x - \pi \left\lfloor \frac x ...
0
votes
2answers
51 views

If the product $fg$ is Riemann integrable then are $f$ and $g$ individually integrable?

If $fg$ is integrable, does this imply that $f$ and $g$ are both integrable too? I don't need a proof, if someone knows please just say (this will help me understand a thing about Taylor's theorem).
0
votes
2answers
48 views

Evaluating the integral : $\int_{1}^{2}\frac{x+\tan x}{x+\sin x}dx$

$Q.$ Evaluate the following integral : $\int_{1}^{2}\frac{x+\tan x}{x+\sin x}dx$. Numerically I found that the answer is roughly $1.000006$ but I am unable to compute using the analytic methods. ...
8
votes
2answers
163 views

Evaluation of a dilogarithmic integral

Problem. Prove that the following dilogarithmic integral has the indicated value: $$\int_{0}^{1}\mathrm{d}x ...
0
votes
0answers
17 views

How to compute using integration the areas of the dodecagons (i.e. twelve-sided polygons) inscribed and circumscribed around a unit circle?

How to compute the areas of the dodecagons (i.e. twelve-sided polygons) inscribed and circumscribed around the unit circle centered at the origin using the methods of the integral calculus?
1
vote
1answer
36 views

Double Integral $\int_{0}^{4} \int_{\sqrt{x}}^{2} \frac{1}{1+y^3} \mathrm{d}y\;\mathrm{d}x$

I am having trouble computing the double integral: $$ \int_{0}^{4} \int_{\sqrt{x}}^{2} \frac{1}{1+y^3} \mathrm{d}y\,\mathrm{d}x $$ I computed the inner integral: $$ \left [ \frac{1}{3}\ln|y + 1| - ...
0
votes
1answer
22 views

What is the value of the unknown parameter so that the given area condition holds?

The graphs of $f(x) \colon= x^2$ and $g(x) \colon= cx^3$, where $c > 0$, intersect at the points $(0,0)$ and $(1/c, 1/c^2)$. What is the value of $c$---and how to compute this value---so that the ...
0
votes
1answer
29 views

Definite Integral theorem validity :- $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds$?

Can we write $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds\tag 1$ ? In other words, is this result valid? If so, could you help me to get the proof it NB :: ...
20
votes
4answers
458 views

How to find ${\large\int}_1^\infty\frac{1-x+\ln x}{x \left(1+x^2\right) \ln^2 x} \mathrm dx$

Please help me to find a closed form for this integral: $$I=\int_1^\infty\frac{1-x+\ln x}{x \left(1+x^2\right) \ln^2 x} \mathrm dx$$
0
votes
2answers
52 views

Definite integral-dot product

I have an integral equation containing dot product $$\int_{0}^{L} \left(\frac{a}{L}.b(s)\right)\mathrm ds\tag 1$$ Data Given a is a constant vector of size 3 b(s) is a varying vector of size 3 " . ...
2
votes
1answer
88 views

How to find $\int_0^1 \frac {\mathrm dx}{\left \lfloor{1-\log_2(1-x)}\right \rfloor}$

We want to evaluate; $$\int_0^1 \frac {\mathrm dx}{\left \lfloor{1-\log_2(1-x)}\right \rfloor}$$ The $\left \lfloor{x}\right \rfloor$ is the floor function. I have made no progress so far.
0
votes
0answers
36 views

Evaluation of $\int_{1}^{\infty} x^{-\frac{5}{3}} \cos \left( \left( x-1 \right) h \right)dx$ with Maple [duplicate]

I have calculated the Integral with the aid of some professors here and I get a problem: $$\int_{1}^{\infty} x^{-\frac{5}{3}} \cos \left( \left( x-1 \right) h \right)dx$$ I have done the Integral ...
-1
votes
1answer
34 views

Explanation of the passage from $\int_{N'}^N dN/N$ to $\ln N-\ln N'$

While going through my text I got stuck in the derivation given in the picture. ($\Omega$ is a constant) I don't know how to get the second step from the first step, also I don't know why ln is ...
0
votes
2answers
33 views

Power series solution to integral equation

Hi guys i'm reading a paper in which the authors have two coupled integral equation for the function $f(x)$ and $g(x)$, in order to solve this problem they employ a power series expansion of these ...
0
votes
1answer
60 views

Quaternion expansion

I have a quaternion equation $ \psi(s)=Pe^{\frac{1}{2}k(s)}\tag 1$ Given conditions and data Here P is a constant unit Quaternion defined for 3D rotation matrix as $(p_1,p_2,p_3,p_4) , p_4\in ...
0
votes
2answers
74 views

Integral of inverse of square root of a quadratic

I haven't taken a course on calculus so far so I don't know what to do. The integral may be wrong. Please tell me which part of it is wrong. $$ q∫_{+a}^{-a}\lim_{c \to g}\frac 1{(b^2+c^2)^{3⁄2}} dc $$ ...
2
votes
4answers
77 views

Does $\int_{-\infty}^\infty \frac{\mathrm dx}{(1+x^2)^\alpha}$ converge?

I'm wondering when the integral $$ \int_{-\infty}^\infty \frac{\mathrm dx}{(1+x^2)^\alpha} $$ converges for the real number $\alpha$.
7
votes
3answers
68 views

How to find the integral $\int_0^{70 \pi} |\cos^{2}x\sin x|\,dx$?

I need help with this problem: $$\int_0^{70 \pi} \left|\cos^{2}\!\left(x\right)\sin\!\left(x\right)\right| dx$$ My friend says it's 140/3 but I don't see how.
3
votes
3answers
75 views

Changing order of integration (multiple integral)

Prove $$ \int_0^a\left( \int_0^x \left( \int_0^y \left( \int_0^z f(u) \, du \right) dz \right) dy \right) dx = \int_0^a \frac {(a-t)^3}{3!} f(t) dt $$ where $a$ is constant. So I began with two ...
7
votes
2answers
159 views

improper integral containing $\sqrt{\cos x-\dfrac{1}{\sqrt 2}}$ in the denominator

How do i find the value of this integral-- $$I=\displaystyle\int_{0}^{\pi/4} \frac{\sec^2 x \ dx}{\sqrt {\cos x-\dfrac{1}{\sqrt 2}}}$$ I came across this integral too in physics.