# Tagged Questions

Questions on the evaluation of definite and indefinite integrals

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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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### Simpler way to compute a definite integral without resorting to partial fractions?

I found the method of partial fractions very laborious to solve this definite integral : $$\int_0^\infty \frac{\sqrt[3]{x}}{1 + x^2}\,dx$$ Is there a simpler way to do this ?
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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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### Will moving differentiation from inside, to outside an integral, change the result?

I'm interested in the potential of such a technique. I got the idea from Moron's answer to this question, which uses the technique of differentiation under the integral. Now, I'd like to consider ...
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### Proof for an integral involving sinc function

I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 dx=\frac{\pi}{2}.$$ What do you think? It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} dx$$ is also ...
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### Calculating the integral $\int_{0}^{\infty} \frac{\cos x}{1+x^2}\mathrm{d}x$ without using complex analysis

Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form? $$\int_0^\infty\frac{\cos\;x}{1+x^2}\mathrm{d}x$$
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### Nonzero $f \in C([0, 1])$ 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 ...
It is changing the coordinate from one coordinate to another. There is an angle and radius on the right side. What is it? And why? I got: $2\mathrm dy\mathrm dx = r(\cos^2\alpha-\sin^2\alpha)\mathrm ... 2answers 283 views ### Methods to evaluate$ \int _{a }^{b }\!{\frac {\ln \left( tx + u \right) }{m{x}^{2}+nx +p}}{dx} $Today I saw a question with an answer that made me rethink of the following question, since it's not the first time I try to find an answer to it. If you look at the answer of Mhenni Benghorbal here ... 6answers 3k views ### Ways to evaluate$\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}$and then do a ... 4answers 2k views ### Prove:$\int_{0}^{\infty} \sin (x^2) dx$converges.$\sin(x^2)$is an example for a function which its limit when$x \to \infty$is not$0$, and still its integral from$0$to$\infty$is finite. I'd like your help with understanding why and a ... 5answers 701 views ### Help solving$\int {\frac{8x^4+15x^3+16x^2+22x+4}{x(x+1)^2(x^2+2)}dx}\displaystyle\int {\frac{8x^4+15x^3+16x^2+22x+4}{x(x+1)^2(x^2+2)}\,\mathrm{d}x}$I used partial fractions, solved$A = 2, C = 3$. $$\frac{A}{x} + \frac{B}{x+1} + \frac{C}{(x+1)^2} ... 3answers 7k views ### Why is the area under a curve the integral? I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself. However when it comes to the area ... 3answers 327 views ### A log improper integral Evaluate :$$\int_0^{\frac{\pi}{2}}\ln ^2\left(\cos ^2x\right)\text{d}x$$I found it can be simplified to$$\int_0^{\frac{\pi}{2}}4\ln ^2\left(\cos x\right)\text{d}x$$I found the exact value in the ... 3answers 534 views ### f uniformly continuous and \int_a^\infty f(x)\,dx converges imply \lim_{x \to \infty} f(x) = 0 Trying to solve f(x) is uniformly continuous in the range of [0, +\infty) and \int_a^\infty f(x)dx converges. I need to prove that:$$\lim \limits_{x \to \infty} f(x) = 0$$Would ... 4answers 329 views ### Evaluating \frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{1-2t\cos\theta +t^2}d\theta I need solve this integral, and I tried various methods of solving and did not get it. The integral is:$$\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{1-2t\cos\theta +t^2}d\theta,$$where t is a ... 5answers 693 views ### Evaluating \int_0^\infty \sin x^2\, dx with real methods? I have seen the Fresnel integral$$\int_0^\infty \sin x^2\, dx = \sqrt{\frac{\pi}{8}}$$evaluated by contour integration and other complex analysis methods, and I have found these methods to be the ... 4answers 906 views ### Evaluate the integral: \int_{0}^{1} \frac{\ln(x+1)}{x^2+1} dx Compute$$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} dx$$2answers 413 views ### Frullani proof integrals How can i prove the theorem of Frullani? I did not even know all the hypothesis that f must satisfy, but I think that this are Let f:\left[ {0,\infty } \right] \to \mathbb R be a a continuously ... 1answer 380 views ### Summing over General Functions of Primes and an Application to Prime \zeta Function Along the lines of thought given here, is it in general possible to substitute a summation over a function f of primes like the following:$$ \sum_{p\le x}f(p)=\int_2^x f(t) d(\pi(t))\tag{1} $$and ... 3answers 190 views ### Indefinite integral of secant cubed I need to calculate the following indefinite integral:$$I=\int \frac{1}{\cos^3(x)}dx$$I know what the result is (from Mathematica):$$I=\tanh^{-1}(\tan(x/2))+(1/2)\sec(x)\tan(x)$$but I don't ... 2answers 725 views ### How to integrate \int e^{-t^{2}} \space dt using introductory calculus methods Earlier today I stumbled across this when I was doing some practice questions for a physics course:$$\int e^{-t^2} \space dt $$To expand, the limits of integration were something like 1 and 4 ... 6answers 484 views ### Integral of \int e^{2x} \sin 3x\, dx I am suppose to use integration by parts but I have no idea what to do for this problem$$\int e^{2x} \sin3x dx$$u = \sin3x dx du = 3\cos3x dv = e^{2x} v = \frac{ e^{2x}}{2} From this I ... 5answers 797 views ### Evaluating \int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x I've been trying to evaluate the following integral from the 2011 Harvard PhD Qualifying Exam. For all n\in\mathbb{N}^+ in general:$$\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$$... 4answers 545 views ### Evaluation of the integral \int_0^1 \frac{\ln(1 - x)}{1 + x}dx How can I evaluate the integral$$\int_0^1 \frac{\ln(1 - x)}{1 + x}dx$$I tried manipulating the known integral$$\int_0^1 \frac{\ln(1 - x)}{x}dx = -\frac{\pi^2}{6}$$but couldn't do anything with ... 4answers 572 views ### How can I compute the integral \int_{0}^{\infty} \frac{dt}{1+t^4}? I have to compute this integral$$\int_{0}^{\infty} \frac{dt}{1+t^4}$$to solve a problem in a homework. I have tried in many ways, but I'm stuck. A search in the web reveals me that it can be do it ... 4answers 1k views ### Computing the integral of \log (\sin x) How to compute the following integral$$\int \log(\sin x)\,\mathrm dx?$$2answers 1k views ### integral with exponential function and logarithm$$ \int_0^{\infty } \frac{\log (x)}{e^x+1} \, dx = -\frac{1}{2} \log ^2(2) $$Anyone an idea on how to prove this? 3answers 734 views ### Name of this identity? \int e^{\alpha x}\cos(\beta x) \space dx = \frac{e^{\alpha x} (\alpha \cos(\beta x)+\beta \sin(\beta x))}{\alpha^2+\beta^2} Again:$$\int e^{\alpha x}\cos(\beta x) \space dx = \frac{e^{\alpha x} (\alpha \cos(\beta x)+\beta \sin(\beta x))}{\alpha^2+\beta^2}$$Also the one for \sin:$$\int e^{\alpha x}\sin(\beta x) ... 2answers 181 views ### Complex Fourier series I need to find the complex Fourier series of this function, and I'm having problems calculating these integers: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ... 3answers 910 views ### Integrate square of the log-sine integral:$\int_0^{\frac{\pi}{2}}\ln^{2}(\sin(x))dx\displaystyle \int_{0}^{\frac{\pi}{2}} \ln(\sin(x))dx=-\frac{\pi}{2}\ln(2)$is an integral that is common. But, how can we show ... 2answers 475 views ###$\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \ dxEvaluate Integral Evaluate $$\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \ dx$$ 1answer 419 views ### A nice log trig integral Show that : $$\int_{0}^{\frac{\pi }{2}}{\frac{{{\ln }^{2}}\cos x{{\ln }^{2}}\sin x}{\cos x\sin x}}\text{d}x=\frac{1}{4}\left( 2\zeta \left( 5 \right)-\zeta \left( 2 \right)\zeta \left( 3 \right) ... 2answers 184 views ### Laplace's method I'm still having a little trouble applying Laplace's method to find the leading asymptotic behavior of an integral. Could someone help me understand this? How about with an example, like: ... 3answers 200 views ### A generalized integral need help I was thinking this integral :$$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$$What I do is use a Reciprocal subsitution, easy to show that: ... 5answers 2k views ### If \int_0^x f \ dm is zero everywhere then f is zero almost everywhere I have been thinking on and off about a problem for some time now. It is inspired by an exam problem which I solved but I wanted to find an alternative solution. The object was to prove that some ... 1answer 190 views ### A few improper integral$$\displaystyle \begin{align*} & \int_{0}^{+\infty }{\frac{\text{d}x}{1+{{x}^{n}}}} \\ & \int_{-\infty }^{+\infty }{\frac{{{x}^{2m}}}{1+{{x}^{2n}}}\text{d}x} \\ & \int_{0}^{+\infty ... 1answer 112 views ### Integral calculus proof Iff(x)$is continuous in$[a,b]$, prove that$ \displaystyle \lim_{n \to \infty} \dfrac{b-a}{n} \displaystyle \sum^n _{k=1} f\left( a + \dfrac{k(b-a)}{n} \right) = \displaystyle \int_a ^ b f(x)dx$... 3answers 67 views ### how to solve an definite integral of floor valute function? the question is, how to proof that this integral: (the integral is definite from 0 to n^2) $$\int_0^{n^2}\lfloor\sqrt{t}\rfloor dt = \frac{1}{6}n(n-1)(4n+1)$$ i'd very much appreciate your help on ... 1answer 337 views ### Insidious exponential integral I hope that someone's up for the challenge; I'm attempting to solve this via computer: \int_{-\pi}^\pi{\displaystyle \frac{e^{i\cdot a\cdot t}(e^{i\cdot b\cdot t}-1)(e^{i\cdot c ... 3answers 5k views ### 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 ... 7answers 2k views ### Lebesgue integral basics I'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's take ... 4answers 1k views ### 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 ... 5answers 1k views ### 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$u=\sin(x)$... 5answers 486 views ### 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)$. EDIT:: I know how to evaluate it, but I am ... 1answer 471 views ### How to show that$\int_0^1 \left(\sqrt[3]{1-x^7} - \sqrt[7]{1-x^3}\right)\;dx = 0$Evaluate the integral: $$\int_0^1 \left(\sqrt[3]{1-x^7} - \sqrt[7]{1-x^3}\right)\;dx$$ The answer is$0,$but I am unable to get it. There is some symmetry I can not see. 1answer 509 views ### Interesting integral formula I looked around and found that integrals of the form $$\int_{0}^{\infty} \frac{x^{m-1}}{a+x^n}, a,m,n \in \mathbb{R}, 0<m<n, 0<a$$ seem to occur very often: Just to give a few examples ... 4answers 631 views ### Notation question: Integrating against a measure Suppose$\mu$is a measure. Is there any difference in meaning between the notation$\int f(x)d\mu(x)$and the notation$\int f(x) \mu(dx)$? Many thanks. 4answers 318 views ### Evaluating$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx\$
How would I go about evaluating this integral? $$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$ What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex ...