Articles of integration

Integral equation solution: $y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t$

Integral equation $$y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t$$ has: a unique solution for $\lambda \neq \frac{4}{\pi +2}$; a unique solution for $\lambda \neq \frac{4}{\pi -2}$; no solution for $\lambda \neq \frac{4}{\pi +2}$, but the corresponding homogeneous equation has a non-trivial solution; or no solution for $\lambda \neq \frac{4}{\pi -2}$, but the corresponding homogeneous equation […]

Integral of Dirac delta function/distribution $\delta(x)$ with upper boundary equal to zero

I would like to find the value of $$\int_{a<0}^0 \delta(x) dx$$ In particular, I would like to know if I can break down the integral $$\int_a^b \delta(x)f(x) dx=\int_a^0 \delta(x)f(x) dx + \int_0^b \delta(x)f(x) dx $$ with $a<0$ and $b>0$ and $f(x)$ a well-behaved function. Is it wrong to break down the integral like this, doing […]

Variable in Feynman Integration

Evaluate: $$I=\int_{0}^{\frac{\pi}{2}} \ln(2468^{2} \cos^2x+990^2 \sin^2x) .dx$$ Now, a friend suggested that to evaluate the integral $I$, I rewrite the integral as $$f(y)=\int_{0}^{\frac{\pi}{2}} \ln(y \cos^2x+ \sin^2x) .dx$$ where $y=\dfrac{2468^2}{990^2}$. $$$$But, how can we do this? $$$$Could somebody please explain how this parametrization was done? Many thanks! $$$$The suggested solution:$$$$ $$f(y) = \int_{0}^{\pi/2} ln( y^{2}cos^{2}x + sin^{2}x)$$ […]

Integration Techniques – Adding values to the numerator.

Suppose you wanted to evaluate the following integral. Where did the 4 come from? I understand that it makes the solution but how would you make an educated guess to put a 4? And how in the future would I solve similar questions?

Why is it not true that $\int_{-\infty}^\infty{\rm} x\,dx=0 \, $ given that x is an odd function?

I was given this to do at my own will and look at why it is divergent, but to be sincere I have no idea on how I can start approaching it. I just need explanation to why the answer is 0. I’m studying this integral: $$\int_{-\infty}^\infty x\,dx=0. $$ Why is it divergent?

What is a counterexample to the converse of this corollary related to the Dominated Convergence Theorem?

Based on Williams’ Probability w/ Martingales: Let $(S, \Sigma, \mu)$ be a measure space. Dominated Convergence Theorem: Suppose $\{f_n\}_{n \in \mathbb{N}}$, $f$ are $\Sigma$-measurable $\forall n \in \mathbb{N}$ s.t. $\lim_{n \to \infty} f_n(s) = f(s) \forall s \in S$ or a.e. in S and $\exists g \in \mathscr{L}^1 (S, \Sigma, \mu)$ s.t. $|f_n(s)| \le g(s) […]

Integral of Bessel function multiplied with sine

I need advice on how to solve the following integral: $$ \int_0^\infty J_0(bx) \sin(ax) dx $$ I’ve seen it referenced, e.g. here on MathSE, so I know the solution is $(a^2-b^2)^{-1/2}$ for $a>b$ and $0$ for $b>a$, but I don’t know how to get there. I have tried to solve it by using the integral […]

Integration with exponential constant

How can I find $$\int_0^\infty \frac{2\left(e^{-t^2} -e^{-t}\right)}{t}\ dt$$ I have been told the answer is $\gamma$, the Euler-Mascheroni constant, but do not understand how this is derived.

Compute $\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$

Compute $$\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$$ The answer is $\pi/2$. The discontinuities at $\pm1$ are removable since the limit exists at those points.

Integral $\int_0^{2\pi}\frac{dx}{2+\cos{x}}$

How do I integrate this? $$\int_0^{2\pi}\frac{dx}{2+\cos{x}}, x\in\mathbb{R}$$ I know the substitution method from real analysis, $t=\tan{\frac{x}{2}}$, but since this problem is in a set of problems about complex integration, I thought there must be another (easier?) way. I tried computing the poles in the complex plane and got $$\text{Re}(z_0)=\pi+2\pi k, k\in\mathbb{Z}; \text{Im}(z_0)=-\log (2\pm\sqrt{3})$$ but what […]