Articles of integration

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.

Integral involving exponential, power and Bessel function

Is there any formula for calculating the following definite integral, including exponential and Bessel function? $$ \int_0^{a}x^{-1} e^{x}I_2(bx)dx$$ Thanks in advance

Any good approximation for this integral?

I am interested in the following integral $$ \mathcal{I}=\int_{-\infty}^\infty\mathop{dz}\left[\frac{1}{\sqrt{a+b}(z^2)^{n/4}}-\frac{1}{\sqrt{a+b\cos^2\theta}(R^2+z^2)^{n/4}}\right], $$ where $R\ll 1$, $n<2$ and $$\cos\theta=\frac{z}{\sqrt{R^2+z^2}}.$$ I was thinking of Taylor Expanding the integrand for $z<R$, and integrating the result from $-R$ to $R$, but the answer seems not good when compared numerically. Any good approximation if not the exact answer would work. Any ideas? […]

for which values of $p$ integral $\int_{\Omega}\frac{1}{|x|^p}$ exists?

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$, containing the origin. My question is as in subject: for which values of $p\in\mathbb{R}$ the integral $\int_{\Omega}\frac{1}{|x|^p}$ exists? It’s easy to find the answer for $n=1$ and then also for $n=2$ and $n=3$, using polar and spherical coordinates respectively, and it seems that the answer should be […]