# Tagged Questions

Questions on the evaluation of definite and indefinite integrals

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13answers
4k views

### Proving $\int_{0}^{+\infty} e^{-x^2} dx = \frac{\sqrt \pi}{2}$

How to prove $$\int_{0}^{+\infty} e^{-x^2} dx = \frac{\sqrt \pi}{2}$$
4answers
707 views

### 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 ?
2answers
897 views

### 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 ...
1answer
1k views

### 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 ...
7answers
3k views

### 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 ...
4answers
1k views

### 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? $$\int_0^\infty\frac{\cos\;x}{1+x^2}\mathrm{d}x$$
5answers
1k views

### 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 ...
3answers
3k views

### $\int_{-\infty}^{+\infty} e^{-x^2} dx$ 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 ...
4answers
2k views

### Explain $\iint \mathrm dx\mathrm dy = \iint r \mathrm d\alpha\mathrm dr$

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

### $\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \ dx$ Evaluate 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

If $f(x)$ 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: \begin{equation} \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 ...
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