Questions about the evaluation of specific definite integrals.
2
votes
0answers
22 views
Setting up proper integral
I can't get a problem to work out the way it is supposed to according to my book.
First, a relevant theorem for the problem:
THEOREM
Suppose $\{V_j; j \in Z\}$ is a multiresolution analysis with ...
3
votes
2answers
74 views
How to solve this integral
Wolfram integrator is unable to solve this:
$$
\int_0^\pi \frac{1}{\sqrt {(a-\cos(t))^2+(b-\sin(t))^2+c^2}}\,\mathrm dt
$$
Any suggestions? Thanks!
2
votes
1answer
31 views
Calculation Of Integral Related To Sequence
Let's evaluate the following integral. Many trials but no success.
$$\int_{-\pi}^{\pi}\dfrac{\sin nx}{(1+\pi^{x})\sin x}dx$$
5
votes
2answers
130 views
$\int_0^1 \sqrt {\tan^{-1}x}\space dx=$?
How to evaluate $$\int_0^1 \sqrt {\tan^{-1}x}\space dx\qquad ?$$ Is there an elementary expression or value for it? (though I know that there is no elementary expression for $\int \sqrt ...
4
votes
2answers
100 views
$\int^{441}_0\frac{\pi\sin \pi \sqrt x}{\sqrt x} dx$
Solve this definite integral:
$$\int^{441}_0\frac{\pi\sin \pi \sqrt x}{\sqrt x} dx$$
1
vote
0answers
34 views
What are some definite integration techniques?
Are there any definite integration techniques which I could learn (calc AB student)? I mean techniques which don't require you to find the anti derivative. Thanks!
2
votes
0answers
66 views
quick integration notation question
What exactly does the base of this integral sign refer to?
$$\int\limits_{[0\to 1]}$$
Is that an arbitrary contour from the point $0$ to the point $1$? (this is complex analysis)
ps. Here's the ...
4
votes
3answers
56 views
Integration using summation
How do you integrate $\sqrt{x}$ from an arbitrary constant $a$ to another $b$ by summation ?
5
votes
0answers
53 views
Closed form for $\int_1^\infty\int_0^1\frac{\mathrm dy\,\mathrm dx}{\sqrt{x^2-1}\sqrt{1-y^2}\sqrt{1-y^2+4\,x^2y^2}}$
Consider the following integral:
$$\mathcal{I}=\int_1^\infty\int_0^1\frac{\mathrm dy\,\mathrm dx}{\sqrt{x^2-1}\sqrt{1-y^2}\sqrt{1-y^2+4\,x^2y^2}}.$$
It can be represented as
...
1
vote
0answers
30 views
Integral of Bessel function of the first kind and exponential function
I would need to know if there's a closed form for the following integral:
$$\int_{0}^{\infty} x^{-1}J_{\frac{1}{2}}(\pi x)J_{\frac{1}{2}}(\pi x)\exp(-b(x-x_0)^2)$$
with $b>0$ and $x_0\in ...
3
votes
1answer
44 views
Using a definite integral, to create a specific recurrence relation.
Hello i have the integral:
$$y_n=\int_0^1\frac{x^n}{x+5}dx$$ where $ n=1,2,3,4,....,\infty$
I need to show that the integral can be represented by the recurrence relation below;
$$y_n= ...
4
votes
0answers
47 views
Good upper bound for $\int_0^1 (1 + 2x)/\sqrt{x + x^3}$
I am trying to obtain an upper and lower estimate for the integral
$$I = \int_0^1 \overbrace{\frac{1}{\sqrt{x^2+1}} \left( \frac{1}{\sqrt{x}} + 2\sqrt{x}\right)}^{\Large f(x)}\,\mathrm{d}x,$$
and an ...
10
votes
1answer
115 views
How do solve this integral $\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx$?
I need to solve the to following integral:
$$\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx.$$
I tried this integral in Mathematica, but it was not able to solve it. ...
1
vote
0answers
34 views
Integral of gaussian and sine/cosine
I really need the solution of two integrals involving exponentials and sine/cosine.
For $n\in \mathbf{N}$ even :
$$\int_{-\infty}^{+\infty}\left(\frac{2\,ax\sin(\pi ...
1
vote
1answer
41 views
Inverse Laplace Transform. Computing the integral.
This question is related to this one, but I'm hereby taking a different approach.
Problem: Solve
$\ddot x+\delta\dot x+\omega_0^2x=\gamma\cos\omega t$.
Find the stationary points and examine their ...
13
votes
3answers
139 views
Find the value of $\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx$
I'm trying to figure out how to evaluate the following:
$$
J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx
$$
I'm tried considering $I(s) = \int_{0}^{\infty}\frac{x^3}{(e^x-1)^s}\,dx\implies ...
2
votes
2answers
47 views
Integrate: $\int_0^{\pi} \log ( 1 - 2 r \cos \theta + r^2)d\theta$
If $r \in \Bbb R$ how to integrate $\displaystyle \int_0^{\pi} \log ( 1 - 2 r \cos \theta + r^2)d\theta$?
I need some hints. Special case, if $r = 1$ then I know the above integral is zero.
Here ...
1
vote
0answers
19 views
Closed form for $k$-th moment
I would like to calculate this $k$-th moment:
$$\int_{-\infty}^{+\infty} \quad x^k\quad \left(i^n\frac{\sin(\pi a x+\frac{n\pi}{2})}{\pi ax+\frac{n\pi}{2}}+(-i)^n\frac{\sin(\pi a ...
13
votes
1answer
93 views
Closed form for $\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm dx$
I encountered this integral in my calculations:
$$\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm ...
5
votes
3answers
115 views
integration by substitution, using $\;t = \tan \left(\frac 12 x\right)$
$\displaystyle\int_0^\frac{\pi}{2}\frac{1}{2-\cos x} \, dx$ using the substitution $t=\tan\frac{1}{2}x$
$x=2\tan^{-1}t$
$\dfrac{dx}{dt}=\dfrac{2}{1+t^2}$
$dx=\dfrac{2}{1+t^2}\,dt$
...
2
votes
0answers
17 views
Area of $ M=\{[x,y] \in R^2; x (x^2+y^2) < x^2-y^2; x>0 \} $
I started out with expressing $y$ in terms of $x$:
$$
\begin{equation}
\sqrt{\frac{x^2-x^3}{x+1}} <y
\end{equation}
$$
Now I integrate over $x \in (0,1)$ since I've graphed the above expression.
...
1
vote
1answer
34 views
$0$-th moment of product of gaussian and sinc function
I would like to calculate the following integrals:
$$\int_{-\infty}^{+\infty} \quad \left(\frac{\sin(\pi a x)}{\pi ax}\right)^2\quad \exp(-bx^2)\,dx$$
$$\int_{-\infty}^{+\infty} \quad ...
6
votes
2answers
100 views
Computation of $\int_0^{\pi} \frac{\sin^n \theta}{(1+x^2-2x \cdot \cos \theta)^{\frac{n}{2}}} \, d\theta$
Show that
$$\begin{align*} \forall x \in [-1,1]: \int_0^{\pi} \frac{\sin^n \theta}{(1+x^2-2x \cdot \cos \theta)^{\frac{n}{2}}} \, d\theta &= c_n \tag{1} \\ \int_0^{\pi} \frac{\sin^{n+2} ...
-7
votes
0answers
70 views
Why Riemann integration is needed? [closed]
What is the necessity of the notion of "Riemann Integration" ? Why is normal definite integral is not good enough ?
10
votes
1answer
102 views
Closed form for $\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.$
I need to find a closed form for these nested definite integrals:
$$I=\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.$$
The inner integral can be ...
5
votes
2answers
61 views
$k$-th moment of product of gaussian and sinc
I would like to calculate the following integrals:
$$\int_{-\infty}^{+\infty} \quad x^k\quad \left(\frac{\sin(\pi a x)}{\pi ax}\right)^2\quad \exp(-bx^2)\,dx$$
$$\int_{-\infty}^{+\infty} \quad ...
0
votes
1answer
22 views
Please help finishing the calculation to find the Entropy of Pareto distribution.
Let $X$ follow Pareto distribution with parameters $\alpha, a, h$. That is, $X\sim Pa(\alpha,a,h)$, where $\alpha>0$ is the shape parameter, $-\infty < a < \infty$ is the location parameter, ...
3
votes
4answers
54 views
What is the definite integral of…
$$\int^L_{-L} x \sin(\frac{\pi nx}{L})$$
I've seen something like this in Fourier theory, but I'm still not sure how to approach this integral. Wolfram Alpha gives me the answer, but no method. ...
1
vote
0answers
29 views
Please help finishing the calculation to prove that ” Pareto distribution & Power distribution has inverse relationship”.
Let X follows Pareto distribution with parameters α, a, h.
that is X~Pa(α,a,h)
Where, α>0 is the shape parameter, -∞< a <∞ is the location parameter, h>0 is the scale parameter.
...
7
votes
3answers
79 views
Integral of a rational function: Proof of $\sqrt{C}\,\int_{0}^{+\infty }{{{y^2}\over{y^2\,C+y^4-2\,y^2+1}}\;\mathrm dy}= {{\pi}\over{2}}$?
I suspect that
$$\sqrt{C}\,\int_{0}^{+\infty }{{{y^2}\over{y^2\,C+y^4-2\,y^2+1}}\;\mathrm dy}=
{{\pi}\over{2}}$$
for $C>0$.
I tried $C=1$, $C=2$, $C=42$, and $C=\frac{1}{1000}$ with Wolfram ...
9
votes
2answers
91 views
$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx$
Is there any closed-form representation for the following integral?
$$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx,$$
where $\mathrm{sech}\,x$ is the hyperbolic secant, ...
1
vote
1answer
41 views
A definite multiple integral
$$\int_0^1\int_\sqrt[3]{x}^1 4\cos(y^4)\,\mathrm dy\,\mathrm dx$$
What I got was
$\sin(1)x+\cos(x^2) dx$ and now I am stuck.
I suddenly froze. Could someone help me? Haven't done calculus for a ...
2
votes
4answers
63 views
Definite integration of a trigonometric function
How to integrate $$\int_0^{\pi/2}\!\dfrac{2a \sin^2 x}{a^2
\sin^2 x +b^2 \cos^2 x}\,dx $$
my first step is $$\frac{2}{a} \int_0^{\pi/2}\!\dfrac{a^2 \sin^2 x}{a^2 +(b^2 - a^2) \cos^2 x}\, dx $$
I ...
4
votes
1answer
60 views
Test of convergence of $\int_{-\infty}^{\infty} \dfrac{x^6+6}{x^8+8}dx$
I am having some trouble with this problem and don't know if I am doing it right:
$$\int_{-\infty}^{\infty} \dfrac{x^6+6}{x^8+8}dx$$
so the steps I have taken so far are, I split it into
...
20
votes
2answers
203 views
$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$
I need to find a closed-form for the following integral. Please give me some ideas how to approach it:
$$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$$
2
votes
1answer
45 views
Integral involving gaussian function
I would like to calculate the following integral:
$$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\quad (x^2+y^2)^k\exp\left(-\dfrac{(x-x_0)^2+(y-y_0)^2}{a^2}\right)\,\mathrm dx\,\mathrm dy$$
Any ...
1
vote
1answer
24 views
Using Triple Integrals to Compute Total Mass
The question I have is as follows:
A rectangular swimming pool with vertical sides that measures 50ft by 150ft has a depth that increases in linearly from the short end (3 feet deep) to the deep end ...
1
vote
2answers
48 views
Complex integral help involving $\sin^{2n}(x)$
Show that $$\int_0^\pi \sin^{2n} \theta d\theta=\dfrac{\pi(2n)!}{(2^n n!)^2} $$
So far I have came up with:
$$\sin^{2n} \theta = \left(\dfrac {z-z^{-1}}{2i} \right)^{2n}$$ and I know I should be ...
2
votes
1answer
52 views
Integral identity with square of Jacobi polynomial
This has stumped me for a while: I have a function $\zeta_k^S(x)$ that can be expressed using Jacobi polynomials $P_k^{(\alpha,\beta)}(x)$:
...
1
vote
3answers
51 views
How to evaluate $\int_1^\infty \frac{1}{x}-\frac{1}{x+1}~dx$
It's a very simple question but it confuses me. How do I evaluate
$$
\int_1^\infty \frac{1}{x}-\frac{1}{x+1}~dx
$$
without splitting? And why can't I split it?
7
votes
1answer
140 views
Prove that $f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$
Let $f:[0,1]\rightarrow \mathbb{R}$ be a differentiable function such that
$$f(x^2)+f(y^2)\le2 f(\sqrt{x y}), \space x,y\ge0 $$
Prove that
$$f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$$
Where should ...
0
votes
1answer
21 views
Show that $\int_0^t\!\!\left(t^2-x^2\right)^n\mathbb{d}x=\frac{\sqrt{\pi}}{2}t^{2n+1}\frac{\Gamma(n+1)}{\Gamma(n+\frac{3}{2})}$
The question asks to prove the identity:
$$\int_0^t\!\!\left(t^2-x^2\right)^n\mathbb{d}x=\frac{\sqrt{\pi}}{2}t^{2n+1}\frac{\Gamma(n+1)}{\Gamma(n+\frac{3}{2})}$$
where $n\in\mathbb{Z}$
I have no idea ...
3
votes
1answer
45 views
$\int_0^1 \Bigl(\mathrm{Li}_2\bigl(\frac{x-1}{x}\bigr)\Bigr)^2 \mathrm{d}x$
This question has some relationship to this integral: Let $\mathrm{Li}_2$ be the dilogarithm. Then, numerically,
$$
\int_0^1 \Bigl(\mathrm{Li}_2\bigl(\frac{x-1}{x}\bigr)\Bigr)^2 \mathrm{d}x
=
...
2
votes
2answers
45 views
How to calculate $\lim_{x\to 1}\int_{0}^{1}\frac{dy}{\sqrt{1-y^{2}}}\frac{y^{3/2}}{\sqrt{x - y}}$ when $x>1$?
Numerically, it looks that the limit is
$$\lim_{x\to 1}\int_{0}^{1}\frac{dy}{\sqrt{1-y^{2}}}\frac{y^{3/2}}{\sqrt{x - y}} = \frac{1}{\sqrt{2}}\log(1 - x) + cte $$,
but I have not been able to ...
0
votes
5answers
85 views
$\int^1_0 \frac{xdx}{x^2+2x+1}$
I need some suggestion how to solve this integral.
$$\int^1_0 \frac{xdx}{x^2+2x+1}$$
I think about to do the following step :
$$\frac{1}{2}\int^1_0\frac{2x+2-2dx}{x^2+2x+1}$$$$ t=x^2+2x+1 \rightarrow ...
8
votes
1answer
92 views
$\int_0^\infty\text{Ci}(x)^3\mathrm dx$
Is there a closed form for this integral:
$$\int_0^\infty\text{Ci}(x)^3\mathrm dx,$$
where $\text{Ci}(x)=-\int_x^\infty\frac{\cos z}{z}\mathrm dz$ is the cosine integral?
16
votes
4answers
202 views
$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx$
Please help me to solve this integral:
$$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx.$$
I managed to calculate an indefinite integral of the left part:
$$\int\frac{\cos x}{\sin ...
16
votes
1answer
161 views
Integrating $\int^{\infty}_0 e^{-x^2}\,dx$ using Feynman's parametrization trick
I stumbled upon this short article on last weekend, it introduces an integral trick that exploits differentiation under the integral sign. On its last page, the author, Mr. Anonymous, left several ...
20
votes
3answers
257 views
+100
An integral involving Fresnel integrals $\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$
I need to calculate the following integral:
$$\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$$
where
$$S(x)=\int_0^x\sin\frac{\pi z^2}{2}\mathrm dz,$$
...
15
votes
1answer
148 views
Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$
Please help me to find a closed form for the following integral:
$$\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx.$$
I was told it could be calculated in a closed form.
