Tagged Questions
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 ...
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
201 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:
$$\int\frac{\cos x}{\sin ...
3
votes
1answer
24 views
Using a particular image to justify a (specific) trig integral equality.
I would like to include the following string of equalities in a paper:
$$\sin ^2(x) + \cos ^2(x) = 1$$
$$\int _0^{\dfrac{\pi}{2}} \sin ^2 (x)dx + \int_0^{\dfrac{\pi}{2}} \cos ^2 (x)dx = ...
1
vote
2answers
111 views
Evaluating a double integral involving exponential of trigonometric functions
I am having trouble evaluating the following double integral:
$$\int\limits_0^\pi\int\limits_0^{2\pi}\exp\left[a\sin\theta\cos\psi+b\sin\theta\sin\psi+c\cos\theta\right]\sin\theta d\theta\, d\psi$$
...
30
votes
2answers
607 views
Prove that $\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$
Prove that
$$\int_0^{\infty} \frac{\sin(2013 x)}{x(\cos x+\cosh x)}dx=\frac{\pi}{4}$$
2
votes
1answer
118 views
Definite double integral of a trigonometric function
I am in trouble in the calculation of the following double integral:
$$\int_0^a d\rho\int_0^{2\pi}d\phi\exp(-ik\rho(\sin(\theta_0)\cos(\phi_0-\phi)+\sin(\theta_1)\cos(\phi_1-\phi)))\rho$$
Many thanks
2
votes
3answers
93 views
Finding integral $\int_{- \pi}^{\pi} \cos(mt) \cos(\lambda t) dt$
I am a little stuck on the following problem:
Prove that:
$$\int_{- \pi}^{\pi} \cos(mt) \cos(\lambda t) dt = -2 \frac{(-1)^{m} \lambda \sin(\pi \lambda)}{m^2 - \lambda ^2}$$
I have used the fact ...
0
votes
1answer
58 views
Is this conjecture about integration of sinusoids on a specific interval correct?
I haven't formally learned integrals, but I was trying to apply what I do know.
Is
$$\int_{a}^{b}m\sin(k(x+j)) dx =0$$
as long as
$b-a\equiv 0 \pmod {\frac{2\pi}{k}-j}$
and $m, k, j \in ...
5
votes
1answer
120 views
How to evaluate $\frac{1}{b^2}\int_0^\infty z^{-2}\exp(-a z)\sin^2(b z)\, \mathrm dz$?
How can I integrate the following:
$$\frac{1}{b^2}\int_0^\infty z^{-2}\exp(-a z)\sin^2(b z)\, \mathrm dz$$
for $a,b>0$? Maple gives a compact result:
$$\frac{1}{b} \tan^{-1}(c) - \frac{1}{ac^2} ...
3
votes
1answer
114 views
evaluation of the integral $\int_{0}^{x} \frac{\cos(ut)}{\sqrt{x-t}}dt $
Can the integral $$\int_{0}^{x} \frac{\cos(ut)}{\sqrt{x-t}}dt $$ be expressed in terms of elemental functions or in terms of the sine and cosine integrals ? if possible i would need a hint thanks.
...
4
votes
1answer
215 views
How can I compute $\int_{-\infty}^\infty f(x)f(y-x)\, \mathrm dx$
If $f(x)=\text{arccot}(x)$ for non-negative $x$ and $0$ otherwise, how can I calculate
$$\int_{-\infty}^\infty f(x)f(y-x)\, \mathrm dx$$
for $y\in\mathbb{R}$?
2
votes
1answer
112 views
Definite integral with a complex number in Euler form
Well... I spent an hour trying to figure out how to go from lhs to rhs:
$$\frac { 1 }{ 2\pi } \int _{ -\infty }^{ +\infty } \phi _{ T }(u)\left( \int _{ k }^{ +\infty } e^{ -iux }dx \right) ...
6
votes
5answers
334 views
A little integration paradox
The following integral can be obtained using the online Wolfram integrator
$$ \int \frac{dx}{1+\cos^2 x} = \frac{\tan^{-1}(\frac{\tan x}{\sqrt{2}})}{\sqrt{2}}$$
Now assume we are performing this ...
4
votes
3answers
349 views
Easiest way to prove that $\int_{\pi/6}^{\pi/2} \sin(2x)^3\cos(3x)^2 \mathrm{d}x=\left(3/4\right)^4$
I have been trying to evaluate this integral a few times. And my best attempt has been to rewrite is as a sum of linear combination of sine and cosine terms. Alas, this takes a couple of handwritten ...
6
votes
6answers
666 views
Integral of $\int_0^{\pi/2} \ (\sin x)^7\ (\cos x)^5 \mathrm{d} x$
I am trying to find this by using integration by parts but I am not sure how to do it.
$$\int_0^{\pi/2} (\sin x)^7 (\cos x)^5 \mathrm{d} x$$
I tried rewriting as
$$\int_0^{\pi/2} \sin x\cdot\ ...
12
votes
0answers
653 views
Computing ${\int\limits_{0}^{\chi}\int\limits_{0}^{\chi}\int\limits_{0}^{2\pi}\sqrt{1- \cdots} \, d\phi \, d\theta_1 \, d\theta_2}$?
This question led me to this integral I can not solve:
$$
...
4
votes
1answer
268 views
Evaluate $\int\limits_0^{\frac{\pi}{2}} \frac{\sin(2nx)\sin(x)}{\cos(x)}\, dx$
How to evaluate
$$ \int\limits_0^{\frac{\pi}{2}} \frac{\sin(2nx)\sin(x)}{\cos(x)}\, dx $$
I don't know how to deal with it.
1
vote
3answers
142 views
Integration: How to Begin? [duplicate]
Possible Duplicate:
Help evaluating $\int \frac{dx}{(x^2 + a^2)^2}$
How to I begin this integration problem?
$\begin{align}\int_{0}^{1} \frac{dx}{{\left(x^2 + 1\right)}^{2}}\end{align}$
...
5
votes
1answer
189 views
Solving $\int_{-\infty}^{\infty}{\frac{1}{(4+x^2)\sqrt{4+x^2}} \space dx}$
I'm trying to solve
$$\int_{-\infty}^{\infty}{\frac{1}{(4+x^2)\sqrt{4+x^2}} \space dx}$$
By substituting $x=2\tan{t}$. I get as far as:
$$\int_{x \space = -\infty}^{x \space = ...
6
votes
1answer
158 views
Integral of difference of sin functions squared is $2\pi$?
Can someone explain why this is true?
$$\int_{0}^{2\pi }\left(\sin nx-\sin kx\right)^2dx=2\pi$$ for natural numbers $n,k$ with $n<k $
2
votes
1answer
449 views
Integral of exponential function with trigonometric identities
I need help in solving the following definite integral. I could not find any example like this
$$\int_{0}^{2\pi}\int_{0}^{d}\exp\!\Big(\frac{-r^2 +2\alpha\; r\cos\theta}{4\;\sigma^2}\Big)r\; dr\; ...
4
votes
4answers
472 views
How to calculate $\int_0^{2\pi} \sqrt{1 - \sin^2 \theta}\;\mathrm d\theta$
How to calculate:
$$ \int_0^{2\pi} \sqrt{1 - \sin^2 \theta}\;\mathrm d\theta $$
3
votes
1answer
384 views
Trig substitution for a triple integral
This problem involves calculating the triple integral of the following fraction, first with respect to $p$:
$$
\int\limits_0^{2\pi} \int\limits_0^\pi \int\limits_0^{2} ...
