# Tagged Questions

Questions on the evaluation of definite and indefinite integrals

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### The Integral that Stumped Feynman?

In "Surely You're Joking, Mr. Feynman!," Nobel-prize winning Physicist Richard Feynman said that he challenged his colleagues to give him an integral that they could evaluate with only complex methods ...
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### Help find hard integrals that evaluate to $59$?

My father and I, on birthday cards, give mathematical equations for each others new age. This year, my father will be turning $59$. I want to try and make a definite integral that equals $59$. So ...
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### Is there an integral that proves $\pi > 333/106$?

The following integral, $$\int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi$$ is clearly positive, which proves that $\pi < 22/7$. Is there a similar integral which proves ...
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### Two curious “identities” on $x^x$,$e$,and $\pi$

A numerical calculation on Mathematica shows that $$I_1=\int_0^1 x^x(1-x)^{1-x}\sin\pi x\,\mathrm dx\approx0.355822$$ and $$I_2=\int_0^1 x^{-x}(1-x)^{x-1}\sin\pi x\,\mathrm dx\approx1.15573$$ A ...
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### $\int_{-\infty}^{+\infty} e^{-x^2} dx$ with complex analysis

Inspired by this recently closed question, I'm curious whether there's a way to do the Gaussian integral using techniques in complex analysis such as contour integrals. I am aware of the calculation ...
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### Ways to evaluate $\int \sec \theta \, d \theta$

The standard approach for showing $\int \sec \theta \, d \theta = \ln |\sec \theta + \tan \theta| + C$ is to multiply by $\frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$ and then do a ...
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### What is so special about $\alpha=-1$ in the integral of $x^\alpha$?

Of course, it is easy to see, that the integral (or the antiderivative) of $f(x) = 1/x$ is $\log(|x|)$ and of course for $\alpha\neq 1$ the antiderivative of $f(x) = x^\alpha$ is ...
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### Proving $\int_{0}^{+\infty} e^{-x^2} dx = \frac{\sqrt \pi}{2}$

How to prove $$\int_{0}^{+\infty} e^{-x^2} dx = \frac{\sqrt \pi}{2}$$
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### Compute $\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$

I've got troubles in computing the below integral: $$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$$ I hope it can be expressed in elementary functions. I've tried simple substitution as $u=\sin(x)$ ...
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### Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
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### How much does symbolic integration mean to mathematics?

(Before reading, I apologize for my poor English ability.) I have enjoyed calculating some symbolic integrals as a hobby, and this has been one of the main source of my interest towards the vast ...
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### Why are gauge integrals not more popular?

A recent answer reminded me of the gauge integral, which you can read about here. It seems like the gauge integral is more general than the Lebesgue integral, e.g. if a function is Lebesgue ...
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### Evaluating $\int P(\sin x, \cos x) \text{d}x$

Suppose $\displaystyle P(x,y)$ a polynomial in the variables $x,y$. For example, $\displaystyle x^4$ or $\displaystyle x^3y^2 + 3xy + 1$. Is there a general method which allows us to evaluate the ...
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### Making trigonometric substitutions rigorous

I've been tutoring some basic calculus, and it made me think about something pretty basic. Let me explain the problem by example: Say we are given the integral $\int \frac{x^2}{\sqrt{1-x^2}}dx$. It ...
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### 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 ...
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### List of interesting integrals for early calculus students

I am teaching Calc 1 right now and I want to give my students more interesting examples of integrals. By interesting, I mean ones that are challenging, not as straightforward (though not extremely ...
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### Generalizing the trick for integrating $\int_{-\infty}^\infty e^{-x^2}dx$?

There is a well-known trick for integrating $\int_{-\infty}^\infty e^{-x^2}dx$, which is to write it as $\sqrt{\int_{-\infty}^\infty e^{-x^2}dx\int_{-\infty}^\infty e^{-y^2}dy}$, which can then be ...
### How to prove that $\int_0^1\left(\sum_{k=n}^\infty {x^k\over k}\right)^2\,dx = \int_0^1 2x^{n-1}\log\left(1+{1\over\sqrt{x}}\right)\,dx.$
American Mathematical Monthly problem 11611 essentially asks you to show that $$\lim_n\ n \int_0^1\left(\sum_{k=n}^\infty {x^k\over k}\right)^2\,dx=2\log(2).\tag1$$ This would follow easily from ...