Articles of integration

Closed form for integral $ \int_0^{\pi} \frac{\sin (m \phi)}{(1 + r \cos \phi)^n} d\phi$

Is there a closed form for $n>0$ integer, $m\neq 0$ integer, and $|r|<1$ real?

How to derive integrals with error function?

How to derive this integral $\int_{-\infty}^{\infty}erf(\lambda x)\mathcal{N}(\mu, \sigma ^2)dx$ and this $\int_{-\infty}^{\infty}(erf(\lambda x)-const)^2\mathcal{N}(\mu, \sigma ^2)dx$ where $erf(x)=\frac{2}{\sqrt\pi}\int_{0}^{x}{e}^{-t^2}dx$ – is the error function $\mathcal{N}(\mu, \sigma ^2)=\frac{1}{\sigma\sqrt{2\pi}}{e}^{-\frac{(x-\mu)^2}{2\sigma^2}}$ – is pdf of the normal distribution From this paper: “…it can be shown by parameter differentiation with respect to mu and then integrating with respect to mu…” they have […]

Can the limit $\lim_{x\to0}\left(\frac{1}{x^5}\int_0^xe^{-t^2}\,dt-\frac{1}{x^4}+\frac{1}{3x^2}\right)$ be calculated?

$$\displaystyle\lim_{x\to0}\left(\frac{1}{x^5}\int_0^xe^{-t^2}\,dt-\frac{1}{x^4}+\frac{1}{3x^2}\right)$$ I have this limit to be calculated. Since the first term takes the form $\frac 00$, I apply the L’Hospital rule. But after that all the terms are taking the form $\frac 10$. So, according to me the limit is $ ∞$. But in my book it is given 1/10. How should I solve […]

Numerical integration over a surface of a sphere

I am integrating a double integral in spherical coordinates over the surface of a sphere in MATLAB numerically. Although I have changed the relative and absolute tolerance I get the feeling that this algorithm never terminates. And when I checked the values of my function that MATLAB had evaluated everything looked fine, no huge oscillations, […]

Partial Fractions with irreducible quadratics in denominator

I am not sure how to begin this question, as we have not really covered it in class: $$\int \frac{3}{x^2 + 4x + 40}\, \mathrm dx$$ Correct me if I am wrong, but it doesn’t seem that the denominator can be reduced further – how might I go about getting started on this?

What exactly is integration?

Consider the function $y=2x$. The graph of this function is here. Next, Consider $\int 2x dx=x^2 + c$. Here is the graph. My question is : What exactly did I do on the straight line $y=2x$ that I landed up with a parabola $y=x^2+c.$

calculate integral $\;2\int_{-2}^{0} \sqrt{8x+16}dx$

I want to calculate the integral $$2\int_{-2}^{0} \sqrt{8x+16}dx$$ The answer is $\;\dfrac {32}{6}\;$ but I don’t know how to get it.

How to solve this double integral involving trig substitution (using tangent function)?

This is a question I came across and I cannot find the answer. By using a substitution involving the tangent function, show that $$\int_0^1\int_0^1\frac{x^2-y^2}{(x^2+y^2)^2}\,dy\,dx=\frac{\pi}{4}$$ My attempt I use trig substitution, by saying $$\tan(\theta)=\frac{y}{x}$$ which means $$x\sec^2(\theta)\,d\theta=dy$$ Also, it should be noted that because of this $$x\sec(\theta)=\sqrt{x^2+y^2}$$ $$x^4\sec^4(\theta)=(x^2+y^2)^2 $$ Thus, when I substitute this information into […]

Trigonometric contour integral

I cannot figure out what I’m doing wrong: $$\int_0^{2\pi} \frac{1}{a+b\sin\theta} d\theta\quad a>b>0$$ $$\int_{|z|=1} \frac{1}{a+\frac{b}{2i}(z-z^{-1})} \frac{dz}{iz}$$ $$\int_{|z|=1} \frac{2i}{2ia+b(z-z^{-1})} \frac{dz}{iz}$$ $$\int_{|z|=1} \frac{2}{2iza+bz^2-b} dz$$ $$2iz_0a+bz_0^2-b=0$$ $$z_0=\frac{-2ia\pm\sqrt{-4^2+4b^2}}{2b}$$ $$z_0=\frac{a\pm\sqrt{a^2 – b^2}}{bi}$$ where only $z_0=\frac{a-\sqrt{a^2 – b^2}}{bi}$ is within $C$. So $Res(z_0) = \frac{-2b}{-2\sqrt{a^2-b^2}}$ So the integral is $\frac{2\pi bi}{\sqrt{a^2-b^2}}$ But this is wrong. The $bi$ should not be there. However […]

Changing the order of integration (for Lebesgue-Stieltjes integral and Riemann integral)

Do the Lebesgue-Stieltjes integral and the Riemann integral have the same rules about the change of order of integration? I mean I know how to deal with Riemann integral, but I’m not sure if I can simply apply the same rules to the Lebesgue-Stieltjes. Thanks. It’s double integration in $\mathbb{R}^2$, by the way.