Articles of integration

Integration of x^(1/2) sinx

My book say that integration of $x^{1/2} \sin x$ is not possible, why is it so? Which functions do not have an anti derivative? Does it mean that they do not have any area under the curve? But that’s not true since the graph says different. (Source: https://www.desmos.com/calculator) When is a function not integrable?

Integration by parts of $\cot x$

I attempted to integrate $\cot x$ by parts by taking $u$ as $\csc x$ and $\dfrac{dv}{dx}$ as $\sin x$. Then: $$\int \cot x\,dx = \int \csc x \cos x\,dx \\ = \sin x \csc x – \int- \sin x \csc x \cot x \, dx \\ = \frac{\sin x}{\sin x} + \int \frac{\sin x}{\sin x}\cot […]

Integrating $ \int_0^2 \int_0^ \sqrt{1-(x-1)^2} \frac{x+y}{x^2+y^2} dy\,dx$ in polar coordinates

I’m having a problem integrating $ \displaystyle\int_0^2 \int_0^ \sqrt{1-(x-1)^2} \frac{x+y}{x^2+y^2} \,dy\,dx$. I drew the graph, and it looks like half a circle on top of the $x$ axis. I tried dividing it into two parts, with Area 1: $0\leq \theta\leq\frac{ \pi }{4} $ with $\csc\theta\leq r\leq 2\cos\theta$ and Area 2: $\frac{ \pi }{4}\leq \theta\leq\frac{ \pi […]

Does $\int_0^\infty \frac{1}{1+(x\sin x)^2}\ dx$ converge?

Does the integral $$\int_0^\infty \frac{1}{1+(x\sin x)^2} \ \, \mathrm{d}x$$ converge? I know that I need to look at: $$\sum_{n=0}^\infty \int_{n\pi}^{(n+1)\pi} \frac{1}{1+(x\sin x)^2}\ \, \mathrm{d}x$$ but not sure how utilize this.

Integrating $\int \dfrac {dx}{\sqrt{4x^{2}+1}}$

$\int \dfrac {dx}{\sqrt{4x^{2}+1}}$ I’ve been up to this one for quite a while already, and have tried several ways to integrate it, using substituion, with trigonometric as well as hyperbolic functions. I know(I think) I’m supposed to obtain: $\dfrac {1}{2}\ln \left| 2\sqrt {x^2 +\dfrac {1}{4}}+2x\right|+c$ However, whatever idea I come up with to try to […]

Multiple integral $\idotsint_V dx_1 \, dx_2 \cdots dx_k$ where $\sum x_i \leq 1$

This question already has an answer here: Volume of an n-simplex [duplicate]

Integrability of $f(t) =\frac{2^{\frac{it+1}{1.5}}}{2^{\frac{it+1}{2}}} \frac{\Gamma \left( \frac{it+1}{1.5} \right) }{\Gamma \frac{ it+1}{2} }$

Can we show that the following function is integrable \begin{align} f(t) =\frac{2^{\frac{it+1}{1.5}}}{2^{\frac{it+1}{2}}} \frac{\Gamma \left( \frac{it+1}{1.5} \right) }{\Gamma \left(\frac{ it+1}{2} \right)}, \end{align} where $t \in \mathbb{R}$ and $i=\sqrt{-1}$. That is can we show that \begin{align} \int_{-\infty}^{\infty} |f(t)| dt<\infty. \end{align} I was wondering if Stirling’s approximation can be used, since this is a complex case? Note if […]

Are discontinuous functions integrable? And integral of every continuous function continuous?

According to me answer of second part is yes as integration simply means area under curve.

How can I prove the following equality

This question already has an answer here: integral $\int_0^\pi \frac{\cos(2t)}{a^2\sin^2(t)+b^2\cos^2(t)}dt$ 3 answers

Integral $\int_{0}^{A}\frac{\exp(-2\pi iwx)}{x-i}dx $

Here $A>0$, $w$-real,$\mathtt{i}$-complex. Mathematica gives the answer: $$\frac{1}{2}e^{2\pi w}(\mathtt{i}\pi+2\Gamma(0,2\pi w)-2\Gamma(0,2(1+\mathtt{i}A)\pi w)+2\ln(-\mathtt{i}+A)+2\ln(w)-2\ln(w+iA)) $$ My questions: 1. How to obtain this results without mathematica? 2. Why does this integral grow exponentially as a function of $w$? Mostly I don’t understand why this integral explodes.The real part is:$$\int_{0}^{A}\frac{x\cos(2\pi wx)+\sin(2\pi wx)}{1+x^{2}}\ {d}x $$ and geometrically I don’t understand why […]