Tagged Questions

Questions about the evaluation of specific definite integrals.

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Finding the limit $\lim\limits_{n \to \infty} \int_1^a \frac{n}{1+x^n} dx$ with $a > 1$

$I_n$ is given by $$I_n=\int_1^a \dfrac{n \ dx}{1+x^n}, \qquad a>1.$$ Find $\lim\limits_{n\to\infty}I_n$.
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Contour Integrals

Evaluate: $\int_C \hat{z} dz$ where $C$ is the straight line from $i$ to $2-i$. $\int_C \frac{dz}{z}$ where $C$ is the straight line from $3$ to $4i$ $\int_C (z-z_0)^{n-1}dz$ for any integer $n$, ...
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Evaluating $\int_0^{2\pi} \sqrt{1+\cos^2(x)}\ dx$ [duplicate]

I want to evaluate this integral $$\int_0^{2\pi} \sqrt{1+\cos^2(x)}\ dx$$ But I cannot find a useful substitution/strategy. Could you please give me a hint? I was thinking proving that this is ...
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About $\int_0^{\pi/2}\arctan(1-\sin^2 (x) \cos^2 (x))dx = \pi \left( \frac{\pi}{4}-\pi \arctan \sqrt{\frac{\sqrt{2}-1}{2}}\right)$

In this question sos440 has mentioned about an integral that he computed: $$\int_0^{\pi/2}\arctan(1-\sin^2 (x) \cos^2 (x))dx = \pi \left( \frac{\pi}{4}- \arctan \sqrt{\frac{\sqrt{2}-1}{2}}\right)$$ ...
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Simplify the integral with error function

$\newcommand{\erf}{\operatorname{erf}}$ I have the following integral and I need to simplify the solution. I have written first two steps. I don't know what is the value of $$\erf(\infty)$$ I ...
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Series sum and integral inaccuracy problem

Well, it happened that math is not my field. Having a vague concept in general I have been struggling for a few hours with the exercise. In general, I want to calculate how much interest a man have ...
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definition of a graph

I have two questions; What is the name of the graph (or circuit) which goes along the outer vertices of existing nodes. What will be the formal definition of that graph. for the ...
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Show $\int_0^b \sqrt{\frac{1 + \cos \theta}{\cos \theta - \cos b}} d\theta= \pi$ for $0 < b < \pi.$

I came across this integral in evaluating the time a particle takes to travel between points on the cycloid. I was able to use substitution to do the integral in the case $b = \pi/2.$ I have not not ...
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Improper integral evaluates to $-\pi^2/12$

$$\int_0^1 \frac{\ln x}{1+x} \mathrm{d}x=-\int_0^1 \frac{\ln(1+x)}{x}\mathrm{d}x=-\frac{\pi^2}{12}$$ Please, help give me proper hints to solve. I was not even able to equate the first two.
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Double integral

I want to evaluate $$\int_{-\infty}^a \int_{-\infty}^b z_1^n z_2^k \phi(z_1,z_2) dz_1 dz_2$$ where $\phi(z_1,z_2)$ is a pdf of bivariate normal distribution and $n,k$ are natural numbers. Are there ...
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The equivalence between Cauchy integral and Riemann integral for bounded functions

Definitions Suppose $P\colon a=x_0<x_1<\dotsb<x_n=b$ is a partition of $[a,b]$. Let $\Delta x_k=x_k-x_{k-1}$ and $\lVert P\rVert$ denotes $\max_{0<k\le n}\Delta x_k$. The Cauchy integral ...