Articles of integration

Multivariable integral over a simplex

Let $p$ be a positive integer, let $B > A >0$ and let $\beta >0 $ and $\beta \neq 1$. With a help of Mathematica (ie using elementary integration and consecutive simplifications) I have shown that : \begin{eqnarray} &&I^{(A,B)}_p := \int\limits_{A\le \xi_0 \le \cdots \le \xi_{p-1} \le B} \prod\limits_{j=0}^p \left(\xi_{j-1} – \xi_j\right) \cdot \prod\limits_{j=0}^{p-1} \frac{d […]

Fourier Cosine Transform (Parseval Identity) for definite integral

I would like to solve the following integral using the Fourier Cosine Transform and its Parseval identitiy. $$I(\gamma,b,\beta)=\int_0^{\infty} \frac{1}{\gamma^2+x^2} e^{-\beta \sqrt{\gamma^2+x^2}} \, \mathrm{cos}(b \, x)\, \mathrm{d}x$$ Therefore I first use the symmetry of the integral to write $$I(\gamma,b,\beta)=\frac{1}{2}\int_{-\infty}^{\infty} \frac{1}{\gamma^2+x^2} e^{-\beta \sqrt{\gamma^2+x^2}+\mathrm{i}\,b\,x} \, \mathrm{d}x$$ and then use the substitution $x=\gamma \, \mathrm{sinh}(t)$ which leads to $$I(\gamma,b,\beta)=\frac{1}{2}\int_{-\infty}^{\infty} […]

Integrating $\int_0^ex^{1/x}\ \mathrm dx$

Compute $$\int_0^ex^{1/x}\;\mathrm dx.$$ There is an analytical anti-derivative found in this answer. How does one compute this? Using the anti-derivative approach we have $$\int x^{1/x}\;\mathrm d x=x + \frac{\log^2x}{2}-\sum^\infty_{n=2}\sum^n_{k=0}\frac{\log^{n-k}x\;}{x^{n-1}(n-k)!(n-1)^{k+1}}+C$$ Now there is a problem for this anti-derivative as it gets near $0$ (apparent from the $\log$ s). I do not know how to prove this […]

$\ln|x|$ vs.$\ln(x)$? When is the $\ln$ antiderivative marked as an absolute value?

One of the answers to the problems I’m doing had straight lines: $$ \ln|y^2-25|$$ versus another problem’s just now: $$ \ln(1+e^r) $$ I know this is probably to do with the absolute value. Is the absolute value marking necessary because #1 was the antiderivative of a squared variable expression that could be either positive or […]

Interpreting Line Integrals with Respect to $x$ or $y$

A line integral (with respect to arc length) can be interpreted geometrically as the area under $f(x,y)$ along $C$ as in the picture. You sum up the areas of all the infinitesimally small ‘rectangles’ formed by $f(x,y)$ and $ds$. What I’m wondering is how do I interpret line integrals with respect to $x$ or $y$ […]

Volume of $T_n=\{x_i\ge0:x_1+\cdots+x_n\le1\}$

Let $T_n=\{x_i\ge0:x_1+\cdots+x_n\le1\}$. I know $T_n$ is tetrahedron. My question: How can I compute the volume of $T_n$ for every $n$?

Nonzero $f \in C()$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$

As the title says, I’m wondering if there is a continuous function such that $f$ is nonzero on $[0, 1]$, and for which $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$. I am trying to solve a problem proving that if (on $C([0, 1])$) $\int_0^1 f(x)x^n dx = 0$ for all $n \geq […]

Evaluate: $\int_0^{\pi} \ln \left( \sin \theta \right) d\theta$

Evaluate: $ \displaystyle \int_0^{\pi} \ln \left( \sin \theta \right) d\theta$ using Gauss Mean Value theorem. Given hint: consider $f(z) = \ln ( 1 +z)$. EDIT:: I know how to evaluate it, but I am looking if I can evaluate it using Gauss MVT. ADDED:: Here is what I have got so far!! $$\ln 2 = […]

How do I integrate the following? $\int{\frac{(1+x^{2})\mathrm dx}{(1-x^{2})\sqrt{1+x^{4}}}}$

$$\int{\frac{1+x^2}{(1-x^2)\sqrt{1+x^4}}}\mathrm dx$$ This was a Calc 2 problem for extra credit (we have done hyperbolic trig functions too, if that helps) and I didn’t get it (don’t think anyone did) — how would you go about it?

Integral of $\sqrt{1-x^2}$ using integration by parts

I was asked to solve this indefinite integral using Integration by parts. $$\int \sqrt{1-x^2} dx$$ I know how to solve if use the substitution $x=\sin(t)$ but I’m looking for the Integration by parts way. any help would be very appreciated.