# Tagged Questions

Questions on the evaluation of definite and indefinite integrals

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### Curve defined by a vector

http://i.stack.imgur.com/tD4Bn.png I'm studying line integrals with a curve as a vector, but I couldn't understand the 'dr' part. First of all: the curve isn't really a curve, it's like some points ...
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### Calculate the limit using definite integrals: $\lim_{n\to\infty}2n\sum_{k=1}^n\frac1{(n+2k)^2}$

Calculate the limit using definite integrals: $\lim_{n\to\infty}2n\sum_{k=1}^n\frac1{(n+2k)^2}$ Well, I started like this: ...
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### Trigonometric function integration: $\int_0^{\pi/2}\frac{dx}{(a^2\cos^2x+b^2 \sin^2x)^2}$

How to integrate $$\int_0^{\pi/2}\dfrac{dx}{(a^2\cos^2x+b^2 \sin^2x)^2}$$ What's the approach to it? Being a high school student , I don't know things like counter integration.(Atleast not taught ...
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### What is $\;\int xe^{-x^2} \,dx\;?$

What is $$\int xe^{-x^2} dx\quad?$$ I used substitution to rewrite it as $$\int -\dfrac{1}{2}e^u\, du$$ but this is too hard for me to evaluate. When I used wolfram alpha for $\int e^{-x^2} dx$ I got ...
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### Extending the result $\int_{0}^{\infty} \left( ( 1 - 2C(x))^{2} + (1-2S(x))^{2} \right) \, dx = \frac{4}{\pi}$

While generalizing this result, I succeeded in proving that for $\alpha > 0$, $\beta < 1$ and $1 < 2\alpha + \beta < 3$, we have \begin{align*} &\int_{0}^{\infty} \left[ \left( ...
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### Calculate the limit: $\lim_{n\to+\infty}\sum_{k=1}^n\frac{\sin{\frac{k\pi}n}}{n}$ Using definite integral between the interval $[0,1]$.

Calculate the limit: $\lim_{n\to+\infty}\sum_{k=1}^n\frac{\sin{\frac{k\pi}n}}{n}$ Using definite integral between the interval $[0,1]$. It seems to me like a Riemann integral definition: ...
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### Integrating: $+ du$ and $\cdot du$

Let's consider the following examples: Evaluate the integral of $6x \sqrt{x^2+1}$ Evaluate the integral of $3x^2 + \sin(x^3-1)$ If I use the substitution method, the first integral gives me the ...
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A follow-up to this question. Which of the following equations is true? Can someone give a proof to the correct equation. Thanks. Please note the prefactors $N$ and $N!$ $$\int_0^A dx_1 \ldots ... 1answer 19 views ### Show relation for integrals Let f \in C^{1}([a,b];\mathbb{R}) and |f'(x)-f'(y)| \le L |x-y| then we have |\int_a^b f(x) dx -f(\frac{a+b}{2})(b-a)| \le L\frac{(b-a)^3}{4}. I have troubles to show this inequality. the ... 1answer 13 views ### Body Volume Rotation Of Shape Question I want to know if Im following the correct step to evaluate the body volume rotation of shape. my function is :$$y=ln(x) and I want to evaluate the body volume rotation of it between $y=0$ and ...
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Im trying to understand how should I evaluate this indefinite integral with this data on the integral : the question is : "Draw the shapes on the plain blocked - by the data lines and evaluate" : 1) ...
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### The sum of the integration of g and $g^{-1}$

Let $g$ be a strictly increasing continuous function mapping $[a,b]$ onto $[A,B]$, and, as usual, let $g^{-1}: [A,B] \to [a,b]$ denote its inverse function. Use geometric insight to visualize the ...