Questions tagged [definite-integrals]
Questions about the evaluation of specific definite integrals.
21,906 questions
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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 ...
7
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5
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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-...
- 5,070
-6
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1
answer
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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
votes
1
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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:
$$
\hat{f}(\omega) = \sqrt{\frac{\omega}{1-e^{-\frac{\omega}{T}}}} e^{- \frac{\omega^2}{4\Lambda^2}}...
- 619
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48
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Picard-Lefshetz method for computing integrals: a simple example
I am trying to use Picard-Lefshetz theory to turn a conditionally convergent integral into absolutely convergent and compute it using the saddle point approximation. The integral is a toy model which ...
- 331
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1
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Help with $\int_{0}^{\pi /4} x^3 (\sqrt{\tan (x)} + \sqrt{\cot (x)}) dx$
I want to evaluate $I=\int_{0}^{\pi /4} x^3 (\sqrt{\tan (x)} + \sqrt{\cot (x)}) dx\tag{0}$
Expressing with $\sin (x)$ and $\cos (x)$:
$$ I = \int_{0}^{\pi /4} x^3 \frac{\sqrt{2}(\sin (x) + \cos (x))}{\...
- 165
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9
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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
$$\int \frac{1}{a \sin ^2x+b \cos ^2x} \...
- 5,070
1
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0
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Definite integral of the two Bessel functions / (x-a)
In my work, an integral of the following type arose:
$$\int_0^\infty dx J_l(a x) J_l(b x) \frac{1}{x - c + i 0}.$$
Here $J$ is the Bessel function of the first kind. Assume that $a$, $b$, and $c$ are ...
4
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1
answer
217
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Real variable method to show that $\int_{-\infty}^\infty \frac{\sinh ax}{\sinh \pi x}\cos bx dx = \frac{\sin a}{\cos a+\cosh b}$?
There are evaluations of the above integral using complex analytic techniques, but is there a real variables way to show it? I've tried the "Feynman differentiation under the integral trick" ...
- 140
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How do I differentiate for a variable ($t$) a definite integral of another variable ($x$)? [closed]
I was reading a pdf for Feynman's trick for integration, and at some point this function is defined:
$f(t):=\int_{0}^{1} \frac{x^{t}+1}{\log(x)}dx$
and later it says:
$f'(t)=\int_{0}^{1} x^{t}dx=\frac{...
- 1
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Triple Integral for Calculating a Volume of a shape rotating around the focus of an ellipse
I am looking at the volume of the shape shown below in the first figure. In particular I'm interested in the area of the element that is rotated around the axis located at the focus of an ellipse. ...
- 491
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2
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On $\int_0^{\frac{\pi}{2}}x\sin\left({\small\frac12}\tan{x}\right)dx=\frac{\pi}{4}\,\frac{\gamma+\delta}{\sqrt{e}}$
Show that:
$$ {\int_0^{\frac{\pi}{2}}x\sin\left({\small\frac12}\tan{x}\right)dx}={\frac{\pi}{4}\,\frac{\gamma+\delta}{\sqrt{e}}}\tag{1} $$
$\,\gamma$: Euler Constant, $\,\delta$: Gompertz Constant
It'...
- 5,232
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4
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Is there a closed form for integral $\int_0^\infty \frac{\ln(1+x^2)}{x^2+2a x+1}dx $ with $|a|<1$
The integral converges since $|a|<1$. In the special case of $a=0$, the integral $$I(a)=\int_0^\infty \frac{\ln(1+x^2)}{x^2+2a x+1}dx
$$
is well-known, which is
$$\int_0^\infty \frac{\ln(1+x^2)}{1+...
- 2,958
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Geometric meaning of double integral when integrating past intersection point
I’m evaluating the double integral
$$\int_{0}^{1/2} \int_{(\sqrt{3})y}^{\sqrt{1-y^{2}}} 1 dx dy$$
The outer limit stops at $y = \frac12$ because the curves
$$x = (\sqrt{3})y$$
$$x = \sqrt{1-y^{2}}$$
...
- 13