Articles of approximate integration

How to use the quaternion derivative

I’m having troubles using the Quaternion derivative. So I have: q(t) = … <– my current attitude The derivative (using w(t) as body rotation rate) is: dq(t)/dt = 1/2 * q(t) * w(t) So how do I use that? When doing stepwise integration (like in a time stepped simulation), I would have expected: q(t+∆t) = […]

Approximate closed form of Integration involving a lot of terms

I tried to solve the following integral using Maple as well as by hand but unable to do so. Can anybody help me in solving the following integral? $$ \int_{0}^{R} D\pi r^2 (D\pi r^2-1)^B 2\pi \lambda \alpha r e^{-\pi r^2(\alpha \lambda – D ln(Y))} dr $$ In the above equation, $D$, $B$, $Y$, $\alpha$, $\lambda$ […]

Real approximation to the maximum using Laplace's method integral

The Laplace’s Method states that under some conditions, it holds that: $ \sqrt{\frac{2\pi}{M(-g”(x_0))}} h(x_0) e^{M g(x_0)} \approx \int_a^b\! h(x) e^{M g(x)}\, dx \text { as } M\to\infty$ Where $g(x_0)$ is the maximum of $g$ and $g”(x_0)$ is the second derivative at that point. (At that point $g”(x_0)<0$. Also $x_0$ is in $(a,b)$.) Inspired in this […]

Integrating $\int_{0}^{\infty} \frac{p^6 dp }{1 + a p^4 + b p^6 } \int_{0}^{\pi}\frac{\sin^5 \theta \,d\theta}{1 + a |p-k|^4 + b |p-k|^6 }$

This is my first question here, so I hope I’m not giving too little/too much information. I need some help calculating (or even approximating) an integral which I’ve been wrestling with for a while. As part of my internship, I need to calculate or even just approximate a power spectrum which boils down to the […]

Average distance to a random point in a rectangle from an arbitrary point

I’m interested in the mean distance between an arbitrary 2D point, $(p, q)$, and a uniformly distributed point inside a rectangle defined by the lower left and upper right vertices $(x_0, y_0)$ and $(x_1, y_1)$, respectively. There is no constraint that $(p,q)$ need lie within the rectangle. I figure the solution is given by $$ […]