# Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

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### Positiveness of sinc function integral

Is there a simple way to see that $$\int_{-a}^0 \int_0^a \frac{\sin(x-y)}{x-y} \, dx \, dy \geq 0$$ for all $a>0$?
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### $\int_0^{\pi} ((\sin{x})^3+(\cos{x})^3)^n dx$ is rational iff $n$ is odd

Define the integral $$I_n = \int_0^{\pi} ((\sin{x})^3+(\cos{x})^3)^n dx$$ for any natural number $n$. I am trying to show that $$I_n\text{ is rational } \iff n \text{ is odd}$$ My first idea was to ...
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### Integral of $e^{\frac{-1}{x(1-x)}}$

The function $e^{\frac{-1}{x(1-x)}}$ is a smooth positive function supported on the compact set $[0,1]$. What is its integral? (According to WolframAlpha, it's approximately 0.007, but I was hoping ...
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### I'm stuck on this integral $\int_{0}^{1} \frac{\ln (x) \ln^{2}(1+x) \ln(1-x)}{x} \ dx$

I was trying to evaluate this famous integral $$\int_{0}^{1} \frac{\ln (x) \ln^{2}(1+x) \ln(1-x)}{x} \ dx$$ Here is my attempt so solve the integral \begin{align} &\int_{0}^{1} \frac{\ln (x) \ln^{...
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### Determine whether $\int_{2}^{\infty}\frac{\cos\left(\ln x\right)}{\left(x+1\right)\ln x}dx$ diverges or converges.

I'm going through my learning material, and one of following question at the end of it was: Determine whether the integral $\int_{2}^{\infty}\frac{\cos\left(\ln x\right)}{\left(x+1\right)\ln x}dx$ ...
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### Separation problem in definite integral of piecewise function

My question is about separation of $\displaystyle \int _0 ^2 f(x) dx$ defined by f(x) = \left\{\begin{aligned} &x^2 ,\ x \in [0,1]\\ &x^4+4 ,\ x \in (1,2] \end{aligned} \right. Of course, we ...
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### Diverging integrals and bounds

My analysis is quite rusty, so I am looking for some help with the following: Suppose we have a function which diverges (logarithmically) at $x = 0$, e.g. $f(x) =\frac{1}{x}$. If we then want to ...
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### Help with Evaluating a Definite Integral Involving Nested Radicals

I'm working on a calculus problem and need help solving the following definite integral: I'm struggling to simplify the integrand or find a substitution that makes the integral easier to evaluate. ...
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### How to evaluate $\int_1^{\infty}\frac{t^2\ln^2 t\ln(t^2-1)}{1+t^6}{\rm d}t$

I was evaluating Evaluate $\displaystyle\int_0^{\infty} x^2\ln(\sinh x)\operatorname{sech}(3 x){\rm d}x .$ On the path of integrating the main function, I am stuck at this integral. I don't know how ...
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### Volume of a Cone using Change of Variables

I am having trouble with the following problem. Let $a,b,c>0$ and let $\Omega$ be the region enclosed by the cone, $\frac{x^2}{a^2}+\frac{y^2}{b^2} =\frac{z^2}{c^2}, 0\leq z\leq c$. I'm tasked to ...
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### Finding a closed form for $\int_0^1 \frac1x \ln\left(\frac{\ln\left(\frac{1-x}{2}\right)}{\ln\left(\frac{x+1}{2}\right)}\right)\, \mathrm{d}x$

I want a closed form for the following integral $$\int_0^1 \frac1x\;\ln\left(\frac {\ln\left(\frac{1-x}{2}\right)}{\ln\left(\frac{x+1}{2}\right)}\right)\, \mathrm{d}x$$ An integration by parts ...
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### Show equivalence of two very long integrals.

I'm trying to show that the following integral $$\int_0^\infty \int_{-\infty}^s f(x \vee s-b) \sqrt{\frac{2}{\pi}}\frac{1}{t^{\frac{3}{2}}}(2s-b)e^{-\frac{(2s-b)^2}{2t}-\mu(b+\frac{\mu t}{2})}db ds$$ ...
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### Product of two improper integals who converge is divergent [closed]

Give an example of two functions $f(x)$ and $g(x)$ such that their integrals both converge respectively, on $[1, \infty]$, but their product integral diverges.
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### Evaluating $4 \sqrt{17}\int_0^1 t e^{-t}dt$ [closed]

I'm not sure how to evaluate this integral. $$4 \sqrt{17}\int_0^1 t e^{-t}dt$$ I know it's supposed to be done by parts, with $u = t$ and $dv = e^{-t}$, but I keep getting $4 \sqrt{17}( -2e^{-1} +1)$ ...
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### Help needed with an integral $\int_0^1 \frac{\ln(x) \ln(1+x) }{1+x^2} dx$.

I am trying to evaluate the integral $$\int_0^1 \frac{\ln(x) \ln(1+x) }{1+x^2} dx$$ Integration by parts gives \begin{eqnarray*} \int_0^1 \frac{\ln(x) \tan^{-1}(x) }{1+x} dx + \int_0^1 \frac{\ln(x) ...
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