Articles of integration

Prove the following equality: $\int_{0}^{\pi} e^{4\cos(t)}\cos(4\sin(t))\;\mathrm{d}t = \pi$

This question already has an answer here: Evaluate $\int_0^{\pi} e^{a\cos(t)}\cos(a\sin t)dt$ 4 answers

a non-monotonic function on with infinitely many points of discontinuity such that the function is bounded & Riemann integrable on .

could any one Give an example of a non-monotonic function on $[0,1]$ with infinitely many points of discontinuity such that the function is bounded & Riemann integrable on $[0,1].$?

Integration and little o notation

Let $f,g : [0, \infty) \rightarrow (0, \infty)$ such that $$f (x) = o(g(x))$$ that is, $ \lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = 0.$ 1) Is it true that $$\int_0^x f(t) dt = o(\int_0^x g(t) dt) \ ?$$ Prove or give counterexample ? 2)If it is not true, what condition should be added to $g(x)$ such […]

$FTC$ problem $\frac d{dx}\int_1^\sqrt xt^tdt$

I stumbled upon a sneaky example while reading about the Fundamental Theorem of Calculus here $$\frac d{dx}\int_1^\sqrt xt^tdt$$ and it makes me question my whole understanding. I know that the FTC says $$\frac d{dx}\int_a^x f(t)dt\,=\, f(x)$$ so I would say that $f(x)=\sqrt x^\sqrt x$ but this is incorrect I should get $$f(x)=\frac 12 x^{\frac {\sqrt […]

How do I solve a double integral with an absolute value?

Given the following integral $$\int_{y=0}^1 \int_{x=0}^1|x-y|(6x^2y) \, dx \, dy$$ how do I change the limits of integration? According to my textbook, it is $$\int_{y=0}^1 \int_{x=y}^{1}(x-y)(6x^2y) \, dx \, dy +\int_{y=0}^1 \int_{x=0}^y (y-x)(6x^2y)\,dx\,dy$$ How did the textbook get the new limits of integration? Can someone explain to me he steps I should use?

How to show the contour integral goes to $0$ of semicircle?

Consider the integral: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ Image taken and modified from: Complex Analysis Solution (Please Read for background information). $R$ is the big radius, $\delta$ is the small radius. We consider $\displaystyle f(z) = \frac{\log^2(z)}{z^2 + 1}$ where $z = x+ iy$ How can we prove: $$\oint_{\Gamma} f(z) dz \to 0 \space \text{when} […]

Integrating $\frac{1}{(x^4 -1)^2}$

How to solve the the following integral? $$\int{\frac{1}{(x^4 -1)^2}}\, dx$$

Question About Orthoganality of Hermite Polynomials

It is known that is $m \ne n$: $$ \int_{-\infty}^{\infty} H_n(x) H_m(x) e^{-x^2}dx = 0 $$ Does this apply for any $f(x)$? $$ \int_{-\infty}^{\infty} H_n(f(x)) H_m(f(x)) e^{f(x)^2} dx = 0 $$

$\lim_n \frac{b-a}{n}\sum_{k=1}^{n}\sup_{x\in}f$, $\lim_n \frac{b-a}{n}\sum_{k=1}^{n}\inf_{x\in}f$ and Darboux integrals

Let us use the notation $\overline{\int_a^b}f(x)dx$ for the Darboux upper integral of $f$ and $\underline{\int_a^b}f(x)dx$ for the lower one. Let us construct a partition of $[a,b]$ into $n$ intervals $[x_{k-1},x_k]$ defined by $x_k=a+k(b-a)/n$ and les us consider the corresponding Darboux sums$$\Delta_n=\frac{b-a}{n}\sum_{k=1}^{n}\sup_{x\in[x_{k-1},x_k]}f(x),\quad \delta_n=\frac{b-a}{n}\sum_{k=1}^{n}\inf_{x\in[x_{k-1},x_k]}f(x).$$ It is clear, by taking the definitions of $\sup$ and $\inf$, and the […]

calculate $\int_0^{\pi}\frac {x}{1+\cos^2x}dx$

Same to the tag calculate $\int_0^{\pi}\frac {x}{1+\cos^2x}dx$. Have no ideas on that. Any suggestion? Many thanks