Articles of integration

Evaluate $\displaystyle I=\int _{ 0 }^{ 1 }{ \ln\bigg(\frac { 1+x }{ 1-x } \bigg)\frac { dx }{ x\sqrt { 1-{ x }^{ 2 } } } }$

Problem: Evaluate: $$\displaystyle I=\int _{ 0 }^{ 1 }{ \ln\bigg(\frac { 1+x }{ 1-x } \bigg)\frac { dx }{ x\sqrt { 1-{ x }^{ 2 } } } }$$ On Lucian Sir’s advice, I substituted $x=\cos(\theta)$. Thus, the Integral becomes $$\int_0^{\pi/2} \ln\bigg(\dfrac{1+\cos(\theta)}{1-\cos(\theta)}\bigg)\dfrac{1}{\cos(\theta)}d\theta$$ $$=\int_0^{\pi/2} \ln\bigg(\cot^2\dfrac{\theta}{2}\bigg)\dfrac{1}{\cos(\theta)}d\theta$$ Unfortunately I’m stuck now. I would be indeed grateful if somebody […]

invariance of integrals for homotopy equivalent spaces

I just wanted to know whether the integral of a closed n-form is invariant if we integrate it over homotopy equivalent spaces. This seems like a generalization of “Homotopy invariance of line integral on manifolds”. Am I right? Something like: $$ \int_{\Sigma} \omega = \int_{\Sigma’} \omega$$ for homotopy equivalent $\Sigma\equiv \Sigma’$ Edit: Ok, I found […]

Defining the multidimensional Riemann Integral as a limit of certain sums

Currently, I am taking a course where we defined the multidimensional Riemann Integral of a map $f:\mathbb{R}^n \to \mathbb{R}$ as the limit $$\lim_{\varepsilon \to 0}\phantom{a}\varepsilon^n \sum_{x \in \mathbb{Z}^n}f(\varepsilon x)$$ If we require that $f$ is continuous and has a compact support, it is guaranteed that the above limit exists. Unfortunately, I do not know any […]

Integral of $\csc(x)$

I’m getting a couple of different answers from different sources, so I’d like to verify something. I misplaced my original notes from my prof, but working from memory I have the following: \begin{align} \int\csc(x)\ dx&=\int\csc(x)\left(\frac{\csc(x)-\cot(x)}{\csc(x)-\cot(x)}\right)\ dx\\ &=\int\frac{\csc^{2}(x)-\csc(x)\cot(x)}{\csc(x)-\cot(x)}\ dx\\ &=\int\frac{1}{u}\ du\\ &=\ln|u|+C\\ &=\ln|\csc(x)-\cot(x)|+C \end{align} This looks proper when I trace it, but wolfram alpha is saying […]

How do I calculate $I = \int\limits_{t_a}^{t_b}{\left(\frac{d{x}}{d{t}}\right)^2 \mathbb dt}$?

$$I = \int\limits_{t_a}^{t_b}{\left(\frac{d{x}}{d{t}}\right)^2 \mathbb dt}$$ $x_a = x(t_a)$ and $x_b = x(t_b)$ I haven’t integrated anything like this since a long time. Lost my powers of integration. How do I calculate $I$?

Integrate $\sin(101x)\cdot \sin^{99}x$

This question already has an answer here: Need help solving – $ \int (\sin 101x) \cdot\sin^{99}x\,dx $ 2 answers

Evaluating a trigonometric integral by means of contour $\int_0^{\pi} \frac{\cos(4\theta)}{1+\cos^2(\theta)} d\theta$

I am studying for a qualifying exam, and this contour integral is getting pretty messy: $\displaystyle I = \int_0^{\pi} \dfrac{\cos(4\theta)}{1+\cos^2(\theta)} d\theta $ I first notice that the integrand is an even function hence $\displaystyle I = \dfrac{1}{2} \int_{-\pi}^{\pi} \dfrac{\cos(4\theta)}{1+\cos^2(\theta)} d\theta $ Then make the substitutions $\cos(n\theta) = \dfrac{e^{in\theta}+e^{-in\theta}}{2}$, and $z=e^{i\theta}$ to obtain: $\displaystyle I = […]

Calculating $\int_{\mathcal{S}}x_1^r \, \mathrm dx_1\ldots \, \mathrm dx_n$

I need help with the calculation of the following integral $$ \int_{\mathcal{S}}x_1^r \, \mathrm dx_1\ldots \, \mathrm dx_n $$ where $r>0$ and $$ \mathcal{S} = \left\{(x_1,\ldots,x_n):a-\epsilon\leq x_1+\ldots+x_n\leq a,\;x_1\ldots,x_n\geq0\right\} $$ for $a>0$ and $a-\epsilon>0$. Thank you

Evaluate the Bessel Function $J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x – \sin x)}\, dx$

I need to evaluate the following definite integral: $$J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x – \sin x)}\, dx$$ I have attempted basic variable substitution and expanding the cosine term, but I have not been able to find an indefinite integral. I believe the best strategy would be to use contour integration, but I am not sure on […]

ODE $d^2y/dx^2 + y/a^2 = u(x)$

Does the following ODE: $$d^2y/dx^2 + y/a^2 = u(x)$$ have a solution? where $u(x)$ is the step function and a is constant.