Articles of integration

Derivative of a function is the equation of the tangent line?

So what exactly is a derivative? Is that the EQUATION of the line tangent to any point on a curve? So there are 2 equations? One for the actual curve, the other for the line tangent to some point on the curve? How can the equation of the tangent line be the same equation throughout […]

Using Parseval's theorem to solve an integral

The question at hand is to use Parseval’s theorem to solve the following integral: $$\int_{-\infty}^{\infty} sinc^4 (kt) dt$$ I understand Parseval’s theorem to be: $$E_g = \int_{-\infty}^{\infty} g^2(t) = \int_{-\infty}^{\infty} |G(f)|^2 df $$ I began by doing the obvious and removing the squared such that: $$g^2(t) = Sinc^4 (kt)$$ $$g(t) = Sinc^2 (kt)$$ Following the […]

Integral of $\int \frac {\sqrt {x^2 – 4}}{x} dx$

I am trying to find $$\int \frac {\sqrt {x^2 – 4}}{x} dx$$ I make $x = 2 \sec\theta$ $$\int \frac {\sqrt {4(\sec^2 \theta – 1)}}{x} dx$$ $$\int \frac {\sqrt {4\tan^2 \theta}}{x} dx$$ $$\int \frac {2\tan \theta}{x} dx$$ From here I am not too sure what to do but I know I shouldn’t have x. $$\int […]

how could we compute this infinite real integral using complex methods?

$\int^{\infty}_{-\infty} \frac{cos(x)}{x^4+1}dx$ I know a similar result, but I’m not sure if I can take it for granted, that $\int^{\infty}_{-\infty} \frac{cos(x)}{x^2+1}dx = \frac{\pi}{e}$ The section in the book is related to Cauchy’s Integral Formulas and Liuville’s th’m, but I’m not sure how to apply these here.

Integrate $x^ {1/2}e^{-x}$ using integration by parts

How to integrate $x^{1/2}e^{-x}$ using integration by parts? Answer should be $\left(-\sqrt{x} e^{-x}+(1/2)\sqrt{\pi} \mbox{erf}(\sqrt{x})\right)+c$

Integrating exponential of exponential function

I would like to find the integral of $\int_0^\infty\exp(-u-\exp(-ku))\,du$ for $k>0$. This is related to the gumbel distribution(, which shows that this integral is one if k=1. However, I would like to know how to integrate this without using the fact that this is a distribution, just so that I can see the method of […]

Integration with a quartic in the denominator $\frac{1}{(x^2-1)^2}$

I had to integrate $$\int\frac{x^2+1}{(x^2-1)^2} dx$$ Well my first approach was to write$\ (x^2+1)$ as $\ (x^2-1)+2$ so as to obtain fractions $$\frac{1}{(x^2-1)} + \frac{2}{(x^2-1)^2}$$ Now I know how to integrate the first part but how to integrate the second part i.e. a quartic (biquadratic) in the denominator? (I got the answer to the original […]

Is sampled absolutely integrable function absolutely summable?

Suppose I have function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that it’s absolutely integrable: $\int_{\mathbb{R}}|f(x)|dx<\infty$. I am sampling function $f(x)$ with some period $T_s$. I am interested whether $$\sum_{k=-\infty}^{k=\infty}|f(kT_s)|<\infty$$ It seems to me that it’s true, but I can’t figure out how to prove that. The reason I ask is that I know that if $f(x)$ is absolutely integrable, […]

Simply this exponential integral

I am trying to simplify of this: $$\int_{0}^{\infty} \frac{1-e^{-x}}{x}e^{-\lambda x}\,dx.$$ Maybe I should separate these equation into two exponential integral function? But it will ended up with infinite minus infinite? please give me some help or advices, thanks!

Behaviour of $\int_0^{\frac{\pi^2}{4}}\exp(x\cos(\sqrt{t}))dt$

I want to analyze the behaviour of $\int_0^{\frac{\pi^2}{4}}\exp(x\cos(\sqrt{t}))dt$, i.e I want to show it behaves like $e^x(\frac{2}{x}+\frac{2}{3x^2}+…)$ as $x\rightarrow \infty$ I started by looking the at the exponent $\cos\sqrt{t}$, taking the derivative and setting it $0$ gives me $0,\pi^2,2^2\pi^2,3^2\pi^2,…$ My guess is to use Lapalce Method now but I do not know how. How can […]