Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

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3
votes
0answers
32 views

How to find $PV\left(\int _0^1\frac{\ln ^3\left(t\right)}{\left(1-t\right)\sqrt{2t-1}}\:dt\right)$

I'm interested in finding in closed-form $$PV\left(\int _0^1\frac{\ln ^3\left(t\right)}{\left(1-t\right)\sqrt{2t-1}}\:dt\right)\approx-0.304615808 + 7.286201516 i$$ but I'm not sure how to evaluate it....
1
vote
1answer
40 views

Let $b>a>0$ , prove $\int_a^b\ln(x)\leq\frac{b^2-a^2}{2}$

I'm confused about this question and I don't know how to achieve this equation. My try : I tried using MVT for integrals so $$\int_a^b\ln(x)=f(c)(b-a)$$ for some $c\in[a,b]$. maximum value for $\ln(x)$...
1
vote
0answers
72 views

How to evaluate $\int_{0}^{1} \cos (4 \pi x) \ln | \zeta(x) | \, dx$

Let $\zeta$ denote Riemann's zeta function. Is it possible to evaluate: $$\int_{0}^{1} \cos (4 \pi x) \ln | \zeta(x) | \, dx$$ or alternatively: $$\int_{0}^{1} x \cos (4 \pi x) \ln | \zeta(x) | \, dx$$...
1
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0answers
14 views

Integrate exponential to the power of the square root term

I have the following integral to compute: \begin{align} &C = \int^\infty_{-\infty} \exp\left(-\sqrt{cb+cx^2} +\sqrt{cb}\right) dx\\ & = \exp\left(\sqrt{cb}\right) \int^\infty_{-\infty} \...
0
votes
1answer
41 views

Are definite integrals always tied to the 'area under the curve'?

Our teacher asked us to solve some integrals and when I asked if it we are solving for the net area or signed area, they said that they are not asking for the area. This got me thinking. Consider the ...
0
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0answers
18 views

Conditional mean calculation

I am trying to solve the following problem. Imagine that a random variable $x$ has a known pdf function $f(x|\sigma)$ ($x$ $\sim$ N(0,$\sigma$)). It's given that $\int_{a}^{\infty}f(x|\sigma)dx=b$, ...
0
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0answers
58 views

Integration question in a math 31 IB (highschool) course

Calculate $$\int_{0}^{\pi/6}\tan^{2}(x)e^{\tan(2x)}\,\mathrm{d}x.$$ This was a question I came by while studying and I have absolutely no idea how to do it. Is it even doable? Maybe with Riemann sums?...
0
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1answer
25 views

Intersection between two intervals

Is the intersection between the following intervals given by $[1, \frac{3}{4}]$? $$[1, \frac{3}{4}]\cap[0, \frac{3}{4}]$$If, for instance, I were defining the bound of a definite integral by the ...
0
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0answers
36 views

How to integrate $\int_0^{\infty} \frac{x^3}{e^{\frac{x-\mu}{kT}} + 1} dx$

So far, I've tried changing the variable to $y = x-\mu$, but doing this makes the lower limit become $-\mu$ instead of $0$: \begin{equation} \int_{-\mu}^{\infty}\frac{y^3 +3y^2\mu+3y\mu^2 + \mu^3}{e^{\...
5
votes
1answer
71 views

Deriving the Integral for Alternating Harmonic Series Partial Sums

The partial sums of the harmonic series (the Harmonic Number, $H_n$) are given by $$H_n=\sum_{k=1}^{n} \frac{1}{k}$$ and the well known integral representation is $$H_n=\int_0^1 \frac{1-x^n}{1-x}\,dx$$...
-3
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0answers
22 views

Proving a square-integrable function with Fourier-series [closed]

We've got a 2$\pi$-periodic function $f:\mathbb{R}\to\mathbb{C}$ that is square-integrable. Here square-integrable means the existence of the right side of the equation. It has the Fourier-...
-4
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0answers
21 views

Evaluate $\iint_{\mathbb R^2}e^{-|x|-|y|}\,dx\,dy$ [closed]

Evaluate $$\iint_{\mathbb R^2}e^{-|x|-|y|}\,dx\,dy$$
2
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1answer
59 views

Compute the area delimited by a curve in implicit form (solution verification)

I have to compute the are delimited by $\sqrt{|x|}+\sqrt{|y|}=\sqrt{a}$ with $a>0$. My idea is, let for instance $a=1$: $$\sqrt{|x|}+\sqrt{|y|}=\sqrt{a}\iff |y|=(1-\sqrt{|x|})^2$$ And since $$ |y| =...
6
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0answers
123 views

About the integral $\int_{0}^{1}\frac{\log(x)}{\sqrt{1+x^{4}}}dx$ and elliptic functions

For a work we need to evaluate the following integral $$\int_{0}^{1}\frac{\log\left(x\right)}{\sqrt{1+x^{4}}}dx=\,-_{3}F_{2}\left(\frac{1}{4},\frac{1}{4},\frac{1}{2};\frac{5}{4},\frac{5}{4};-1\right).\...
0
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1answer
28 views

Why $\frac{1}{-j(\omega-\omega_0)}e^{-j(\omega-\omega_0)t}\bigg |_{-\infty}^{\infty}=0$ when $\omega \neq \omega_0$?

In the class, my professor said the following $$\int_{-\infty}^{\infty} e^{-j(\omega-\omega_0)t} dt=\frac{1}{-j(\omega-\omega_0)}e^{-j(\omega-\omega_0)t}\bigg |_{-\infty}^{\infty}=0,$$ by Euler's ...
1
vote
1answer
77 views

Prove $\int_{0}^{\infty} \frac{x^{p-1}}{1+x} dx = \frac{\pi}{\sin (p \pi)}$ without using residue theorem or Beta function

I need to prove that $$\int \limits_{0}^{\infty} \frac{x^{p-1}}{1+x} dx = \frac{\pi}{\sin (p \pi)}$$ for $0<p<1$. I know how to do it using residue theorem and using the Beta function integral ...
2
votes
0answers
108 views

Challenging integral $I=\int_0^{\pi/2}x^2\frac{\ln(\sin x)}{\cos x}dx$

My friend offered to solve this integral. $$I=\int_0^{\pi/2}x^2\frac{\ln(\sin x)}{\cos x}dx=\frac{\pi^4}{32}-{4G^2} $$ Where G is the Catalan's constant. $$I=\int _0^{\infty }\frac{\arctan ^2\left(u\...
2
votes
1answer
111 views

Integration $\int_{\frac{1}{10}}^{\frac32}\frac1k da$

This is the question I tried to solve and got answer as wrong 3 times. $\color{blue}{\text{I feel the correct answer should be 0.23 but it says correct is 0.557}}$. My try: I split the integral at $a=\...
0
votes
2answers
57 views

Number of functions satisfying $\int_{0}^1xf(x)dx=\frac{1}{3}+\frac{1}{4}\int_{0}^1(f(x))^2dx$

The number of continuous functions $f:[0,1]\to R$ that satisfy $$\int_{0}^1xf(x)dx=\frac{1}{3}+\frac{1}{4}\int_{0}^1(f(x))^2dx$$ is (A) $0$ (B) $1$ (C) $2$ (D) infinity My Attempt: $$\int_{0}^1xf(x)dx=...
5
votes
3answers
152 views

taking an integral without trigonometric substitution

I needed to evaluate $$\int_{-\infty}^\infty\frac{1}{x^2+a^2}dx$$ I looked around and found it's solved by trigonometric substitution $x = a\tan\theta$ and the answer is $\frac{\pi}{a}$. I understand ...
1
vote
2answers
93 views

How to integrate $\int_0^{\infty}\frac{1}{(1+x)(1+x^2)}$

What is the method of integrating the following: $$\int_0^{\infty}\frac{1}{(1+x)(1+x^2)}$$ I tried doing it via using partial fractions and deduce that: $$\frac{\ln(x+1)}{2} - \frac{\ln(x^2+1)}{4} + \...
2
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0answers
46 views

An Inequality concerning double integral

Prove $$ \frac{\pi}{8}\left(1-\cos\frac{2}{\pi}\right)\le \iint_D \sin (x^2)\cos (y^2) dxdy \le \frac{\pi}{8}(1-\cos 1),$$where the integrating region $D$ is enclosed by $x=0, y=0$ and $x+y=1$. We ...
-1
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1answer
39 views

how to show using integration $\int_1^M x^{n+1}e^{−x}$ by part that for any real M> 1?

After having the \begin{align*} \int_1^M &x^{n+1}e^{−x} dx \\ &= −M^{n+1}/e^{M}+1/e+\int_1^M (n+1)x^{n} e^{−x}dx \\ &= [x^{n+1} e^{x}]-\int_1^M e^{-x} (n+1)x^{n} e^{−x}dx \\ &= M^{n+...
0
votes
3answers
89 views

How to solve $\int_{-\infty}^0\frac{e^x\sin x}{x}dx$?

$z=x+iy$, solve $I=\int_0^1\frac{\sin(\ln(Re(z)))}{\ln(Re(z))}d(Re(z))$ $Re(z)=x$, so, I=$\int_0^1\frac{\sin(\ln x)}{\ln x}dx$. Put $\ln x=t$, so $\frac{dx}{x}=dt\implies dx=e^tdt$, so $I=\int_{-\...
1
vote
0answers
43 views

Lebesgue Dominated Convergence theorem question

Prove $$\lim_{r\to 1^-}\int_{0}^{2\pi}\frac{d\theta}{(1+r^2\cos2\theta)^{\frac{1}{3}}}= \int_{0}^{2\pi}\frac{d\theta}{(1+\cos2\theta)^{\frac{1}{3}}} $$ Answer by Parcly By symmetry considerations ...
0
votes
0answers
32 views

Show that $\lim_{n \to \infty} \int_0^1 [f(x)]^n \text{d}x=\infty \implies \exists x_0 \in [0,1] \ \text{s.t} \ f(x_0)>1$

Let $f:[0,1] \to \mathbb{R}$ be a continuous and nonnegative function. Show that $$\lim_{n \to \infty} \int_0^1 [f(x)]^n \text{d}x=\infty \iff\exists x_0 \in [0,1] \ \text{s.t} \ f(x_0)>1$$ I'm ...
15
votes
3answers
308 views

Find $\int_0^1 \dfrac{f(x)}{\sqrt{1+x^2}}$

Let $f(x)$ be continuous on $[0;1]$, with $f(0) = 0; f(1) = 1$ and $$\int_0^1 [f'(x)]^2 \sqrt{1+x^2} \,dx = \dfrac{1}{\ln\left(1+\sqrt{2}\right)}$$ Find ${\displaystyle \int_0^1} \dfrac{f(x)}{\sqrt{1+...
6
votes
1answer
134 views

Ways to prove a relation featuring $\int _0^{\infty }\frac{\arctan ^2\left(x\right)\ln ^2\left(1+x^2\right)}{x^2}\:dx$

From a previous post the following integral arised: $$\int _0^{\infty }\frac{\arctan ^2\left(x\right)\ln ^2\left(1+x^2\right)}{x^2}\:dx$$ And there user178256 linked us to this question where the user ...
0
votes
1answer
48 views

How do I calculate the value of an infinite series of integrals?

So here's my problem. I'm trying to run a simulation of a mass moving through two other fixed masses (with gravity). Let's call the moving mass mass 1, and the stationary masses mass 2 and mass 3. ...
0
votes
1answer
67 views

how can I show that the integral is convergent $\int_1^\infty\ e^{-x} dx$ and that it is equal to $1/e$

After having the $0<e^{−x}$ for all x≥1,and we see $$\int_1^\infty\ e^{−x} dx$$ so $I_0$ converge, $$dx=1/e$$ Moreover, for any natural number n non-zero, $$I_n= \int_1^n x^{n}e^{−x} dx$$ Using ...
1
vote
3answers
89 views

Evaluating $\int_0^B \frac{1}{A^2 + (B^2-x^2)^2}\,dx$

I am trying to solve the following integral: $$\int_0^B \frac{1}{A^2 + (B^2-x^2)^2}\,dx$$ However the polynomial have imaginary roots: $$-(-i A + B^2)^{1/2},\; (-i A + B^2)^{1/2},\; -(i A + B^2)^{1/2},...
2
votes
3answers
73 views

Integral of the form $\int_\Omega |\mathbf{r}_1 - \mathbf{r}_2| \exp\left(-\frac{1}{2}(r_1^2+r_2^2)\right)\mathrm{d}\mathbf{r}_2$

I am trying to derive the density of Hooke's atom. I know this is a physics model, but I think my problem is more in the realm of mathematics than physics. Let me give a quick description of the model ...
0
votes
2answers
32 views

find the volume of indicated solid.

under $$x^2+y^2+z^2=6$$ and above $$z=x^2+y^2$$. i don't know how to continue
0
votes
1answer
51 views

When can you switch the order of differentiation and integration?

Suppose I have some functions $f(t), g(f)$ and some constants $a, b$ and I'm computing the integral $$\int_{a}^{b} \left(\frac{d}{dt}g(f(t))\right) df$$ Then would it be valid to compute: $$\left[\...
2
votes
1answer
42 views

Property about odd function

Let $y: \mathbb{R} \longrightarrow \mathbb{R}$ be an odd and continuous function and $T>0$. Suppose that there exists $\varphi: \mathbb{R} \longrightarrow \mathbb{R}$ a continuous function such ...
6
votes
2answers
112 views

Find the integral $\int_0^\infty \mu^x / \Gamma(x + 1) dx$ [duplicate]

Basically, I'm looking for advice on how I could find the value of $$\int_0^\infty \frac{\mu^x}{\Gamma(x + 1)}dx $$ where $\mu > 0$ is an arbitrary positive constant. Based on the infinite series, ...
-1
votes
1answer
35 views

Integral of double sin

Consider $n\in\mathbb{N}$ and $l\in\mathbb{N}$. How do I calculate the integral that follows? $$\int_0^L x\sin\left(\frac{\pi n x}{L}\right)\sin\left(\frac{\pi l x}{L}\right) dx$$
2
votes
1answer
67 views

Proving that $-\frac{\pi}{2}\le\int_{-1}^{1}\arctan (x)dx\le \frac{\pi}{2}$

I have the following question: Prove that: $$-\frac{\pi}{2}\le\int_{-1}^{1}\arctan (x)dx\le \frac{\pi}{2}.$$ I know that the function is odd and therefore, the given integral is 0, and the ...
2
votes
2answers
36 views

Getting different answers for a definite Integral using different approaches.

I am trying to solve this definite integral problem. $$\int_0^\pi \frac{dx}{1+\cos^{2} x}$$ dividing numerator and denomenator by $\cos^{2} x$, the integrand becomes, $$\int_0^\pi \frac{\sec^{2} x \ ...
3
votes
3answers
107 views

Let $\int_0^2 f\left(x\right) dx = a+\frac{b}{\log 2}$. Find $a,b$

Let $f$ be a real-valued continuous function on $\mathbb{R}$ such that $2^{f\left(x\right)}+f\left(x\right)=x+1$ for all $x\in \mathbb{R}$. Assume that $\int_0^2 f\left(x\right) dx = a+\dfrac{b}{\log ...
1
vote
3answers
74 views

If $dF=f(x)dx$, should I write $\int_a^b f(x)dx$ as $\int_a^bdF$? or as $\int_{F(a)}^{F(b)}dF$?

What is a proper way to change the differential of an integral? For example suppose we have the following integral: $$\int_1^2 2x dx$$ which equals 3. But we know that $2x dx = d(x^2)$. Should I write:...
1
vote
0answers
40 views

Give examples when Mean value theorem doesn't work when $f$ is discontinuous or $g$ is changing signs

I'm really confused on this question , I have tried over 8 examples and didn't find anything that fits. I need to find 2 functions that : (1) - Using the MVT for integrals - Find discontinuous ...
22
votes
2answers
450 views
+50

Integral $\int_0^\infty\frac{\log(1+\cos x)}{1+e^x}\,dx=0$

In this question, the result $$\int_0^\infty\frac{\log\cos^2x}{1+e^{2x}}\,dx=-\frac{\log^22}2$$ was shown by writing $\log\cos^2x=2\log\cos x$. The OP of the linked question attempted the integral as ...
1
vote
2answers
41 views

Limit of a definite nonelementary integral

I have to prove that $\lim_{n\to \infty}\int_0^{1/2}x^ne^{2x-1}\;dx=0$ without calculating the primitive. I have already proved that it can be defined as a recurrence sequence: $$y_1=\frac{1}{4e},\...
0
votes
1answer
35 views

Find the area bounded by $y = \sin^3(x) + \cos^3(x)$ and $y = 0$ if $x \in [-\pi/4, 3\pi/4]$

It took me hours to solve this. I had to split the given segment into three parts- $[-\pi/4, 0], [0, \pi/2], [\pi/2, 3\pi/4]$. Then I evaluated the definite integrals for $sin^3x$ and $cos^3x$ on each ...
8
votes
1answer
219 views

How to calculate the integral $\int_0^{+\infty}\frac{\ln(\cos^2x)}{1+e^{2x}}dx$

Calculate the integral $$\int_0^{+\infty}\frac{\ln(\cos^2x)}{1+e^{2x}}\,dx.$$ I tried $$\displaystyle\ln(\cos^2x)=\ln\left(\frac{\cos2x+1}{2}\right)=\ln(1+\cos2x)-\ln2.$$ It's easy to get the result ...
2
votes
3answers
66 views

Showing convergence of integral sequence

Suppose we have the integral $I_n = \frac{q^n}{n!}\int\limits_0^\pi x^n(\pi-x)^n \hspace{2pt} dx$ for some positive integer $q$. I'd like to prove that the sequence $\{I_n\}$ converges to zero. This ...
5
votes
2answers
61 views

Problem manually evaluating definite integral

I seem to be quite stuck when trying to normalize a probability density given as $$p(x|\omega_i)\propto e^{-\frac{|x-a_i|}{b_i}}$$ with $a_i\in R$ and $b_i\in R^+$. Although I was able to "...
5
votes
3answers
85 views

Integral inequality in proof of irrationality of $\pi$

I'm trying to follow along with my textbook's proof of $\pi$ being irrational. One step in the proof (which my book takes for granted) confuses me. It states that, for each positive integer $n$, $$0 &...
-5
votes
0answers
24 views

If the antiderivative of f is differentiable, then is f integrable? [closed]

If the antiderivative of f is differentiable, then is f integrable?

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