# Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

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### 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....
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### 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 $\ln(x)$...
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### 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: $$\int_{0}^{1} x \cos (4 \pi x) \ln | \zeta(x) | \, dx$$...
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### 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} \...
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### 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 ...
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### 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 $\int_{a}^{\infty}f(x|\sigma)dx=b$, ...
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### 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?...
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### 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 ...
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### 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^{\...
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### 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 $$H_n=\int_0^1 \frac{1-x^n}{1-x}\,dx$$...
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### 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-...
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### Evaluate $\iint_{\mathbb R^2}e^{-|x|-|y|}\,dx\,dy$ [closed]

Evaluate $$\iint_{\mathbb R^2}e^{-|x|-|y|}\,dx\,dy$$
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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