Questions tagged [definite-integrals]
Questions about the evaluation of specific definite integrals.
21,913 questions
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Infinite Sums Containing Alternating Euler Sum for Odd Powers
When I was evaluating this monstrous integral $$
\int_0^{\frac{\pi}{2}} x^3 \ln^2 \left(\sin x\right) \, \mathrm{d}x
$$
I managed to reduce it using the fact that $$
\ln^2\left(\sin x\right) = \frac{\...
- 65
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A very weird integral equation
I tried to come up with an integral equation for fun, and made this creature:
$$
f(x)-\int_x^{2x}f(t)dt=0
$$
so I followed these steps:
$$
\begin{aligned}
&f(x)+\int_x^{2x}f(t)dt=0\\
\Rightarrow\...
- 61
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Generalization of a double integral involving the logarithmic mean
I recently came across the following two integral identities relating to the Riemann Zeta function:
$$
\int_0^1 \int_0^1 \frac{\log(1/x) - \log(1/y)}{\log(\log(1/x)) - \log(\log(1/y))} \,dx\,dy = \...
- 436
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Asymptotics of the integral $\int_0^1 \frac{1-(1-t)^n}{t} dt$
Question
Find an asymptotic formula for the integral $$I_n = \int_0^1 \frac{1-(1-t)^n}{t} dt$$ as $n\to \infty$.
source: this question (the case $b=2$).
I thought about Laplace's method but the ...
- 12.7k
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On $\int_{0}^{\frac{\pi}{2}} \sin(x)^{-\sin(x)}$, and other similar "Sophomore's dream"-esque integrals.
This is a problem I have been tussling with for a while, with varying progress through the years. I think I am finally stuck, or at least cannot move forward with knowing significantly more. As such, ...
- 41
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Closed form for $\int_{0}^\infty e^{-ax^2}\operatorname{erf}(px)\operatorname{erf}(qx)\operatorname{erf}(rx)\operatorname{erf}(sx)\,dx$?
Is there a closed form for this integral?
$$\int_{0}^\infty e^{-ax^2}\operatorname{erf}(px)\operatorname{erf}(qx)\operatorname{erf}(rx)\operatorname{erf}(sx)\,dx$$
Recall the definition of the error ...
- 85.5k
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Prove that $2\int_0^1 \frac{\operatorname{Li}_2(x(1-x))}{1-x+x^2}\mathrm{d}x=\int_0^1 \frac{\ln x \ln(1-x)}{1-x+x^2}\mathrm{d}x$
I'd like to prove that
$$2\int_0^1 \frac{\operatorname{Li}_2(x(1-x))}{1-x+x^2}\mathrm{d}x=\int_0^1 \frac{\ln x \ln(1-x)}{1-x+x^2}\mathrm{d}x.$$
Ok, someone said that this holds, but I tried really ...
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Getting different answers for integration problem: $\int_0^2 x d( \{x\} )$
My teacher used integration by parts to solve the problem like so:
$$\int_0^2 xd(\{x\})
=[x\{x\}]_0^2-\int_0^2 \{x\}dx\\
=0-\int_0^1 xdx-\int_1^2 (x-1)dx$$
which comes out to -1. But when I was ...
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Prove $\int_0^af(x)\mathrm dx\ge a\int_0^1f(x)\mathrm dx$ for decreasing function
Let $f$ be a decreasing function on $[0,1]$ and $a\in(0,1)$. Prove that
$$\int_0^af(x)\mathrm dx\ge a\int_0^1f(x)\mathrm dx.$$
This would be quite obvious if $f$ were continuous. But for non-...
- 5,082
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Is this a probability density function? [closed]
Given $fx(x) = \{ \frac{1}{\pi} \; \text{for} \; x_1 + x_2 \le 1$
I am required to state if the function represents a density function and prove why. I know that to prove it I must check that $f(x) \...
5
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The Multiplicative Role of $dx$ in Indefinite and Definite Integrals: A Comparison with Derivative Notation
I understand from prior discussions (e.g., What does the $dx$ mean in the notation for the indefinite integral?) that $dx$ in $\int f(x) \, dx$ serves as more than mere notation for the variable of ...
- 321
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Inverse Fourier transform for the square root of the Ohmic bath spectral function
I think this is a bit hopeless but let me ask just in case. Consider the real and positive function:
$$
\hat{f}(\omega) = \sqrt{\frac{\omega}{1-e^{-\frac{\omega}{T}}}} e^{- \frac{\omega^2}{4\Lambda^2}}...
- 619
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Picard-Lefshetz method for computing integrals: a simple example
I am trying to use Picard-Lefshetz theory to turn a conditionally convergent integral into absolutely convergent and compute it using the saddle point approximation. The integral is a toy model which ...
- 331
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Help with $\int_{0}^{\pi /4} x^3 (\sqrt{\tan (x)} + \sqrt{\cot (x)}) dx$
I want to evaluate $I=\int_{0}^{\pi /4} x^3 (\sqrt{\tan (x)} + \sqrt{\cot (x)}) dx\tag{0}$
Expressing with $\sin (x)$ and $\cos (x)$:
$$ I = \int_{0}^{\pi /4} x^3 \frac{\sqrt{2}(\sin (x) + \cos (x))}{\...
- 165
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Evaluate $\int_0^{\frac\pi2}\frac{\mathrm dx}{a\sin ^2x+b\cos ^2x}$ without using its antiderivative
Find the value of
$$\int_0^{\frac\pi2} \frac{1}{a \sin ^2x+b \cos ^2x} \, \mathrm dx,$$
where $a$, $b>0$.
The corresponding indefinite integral evaluates to
$$\int \frac{1}{a \sin ^2x+b \cos ^2x} \...
- 5,082