# Tagged Questions

Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

3k views

3k views

### Integral Contest

Before you answer this OP, please read all the terms and conditions below. Thank you... Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for ...
1k views

### Prove $\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}\mathrm dx=\frac{\pi^2}8-\frac12$

How can I prove the following identity? $$\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}\mathrm dx=\frac{\pi^2}8-\frac12$$
3k views

1k views

### 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 ...
509 views

### Mathematical meaning of certain integrals in physics

While studying on texts of physics I notice that differentiation under the integral sign is usually introduced without any comment on the conditions permitting to do so. In that case, I take care of ...
1k views

### A Putnam Integral $\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)} + \sqrt{\ln(x+3)}}.$

This is a Putnam Problem that I have been trying to solve (on and off) for two years, but I have failed. I am in Calculus BC. This problem comes from the book "Calculus Eighth Edition by Larson, ...
$$\int_0^{\frac{\pi}{2}}\arctan\left(\sin x\right)dx$$ I try to solve it, but failed. Who can help me to find it? I encountered this integral when trying to solve $\displaystyle{\int_0^\pi\frac{x\... 6answers 826 views ### show that$\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$show that $$\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$$ using different ways thanks for all 4answers 3k views ### When is an elliptic integral expressible in terms of elementary functions? After seeing this recent question asking how to calculate the following integral $$\int \frac{1 + x^2}{(1 - x^2) \sqrt{1 + x^4}} \, dx$$ and some of the comments that suggested that it was an ... 4answers 6k views ### Using Integration By Parts results in 0 = 1 I've run into a strange situation while trying to apply Integration By Parts, and I can't seem to come up with an explanation. I start with the following equation: $$\int \frac{1}{f} \frac{df}{dx} ... 3answers 729 views ### Compute \int_0^{\pi/2}\frac{\sin 2013x }{\sin x} \ dx\space How would you approach$$\int_0^{\pi/2}\frac{\sin 2013x }{\sin x} \ dx\space?$$The way I see here involves Dirichlet kernel. I wonder what else can we do, maybe some easy/elementary approaching ... 5answers 51k views ### Is there a rule of integration that corresponds to the quotient rule? When teaching the integration method of u-substitution, I like to emphasize its connection with the chain rule of integration. Likewise, the intimate connection between the product rule of derivatives ... 3answers 536 views ### Evaluating a sum involving binomial coefficient in denominator I came across the following sum:$$\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}\frac{4^k}{{2k \choose k}}$$I thought that this can be evaluated using the expansion of \dfrac{\sin^{-1}x}{\sqrt{1-x^... 4answers 715 views ### Volume of T_n=\{x_i\ge0:x_1+\cdots+x_n\le1\} Let T_n=\{x_i\ge0:x_1+\cdots+x_n\le1\}. I know T_n is tetrahedron. My question: How can I compute the volume of T_n for every n? 1answer 169 views ### To find the minimum of \int_0^1 (f''(x))^2dx I was trying to solve a question of an entrance exam. I am completely stuck in the problem. I am not able to find idea how to proceed. Please help me. Let A be the set of twice continuously ... 2answers 4k views ### List of functions not integrable in elementary terms When teaching integration to beginning calculus students I always tell them that some integrals are "impossible" (with a bit of expansion on what that actually means). However I must admit that the ... 3answers 630 views ### Find functions family satisfying \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1) I wonder what kind of functions satisfy$$ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$$I suppose all functions must be continuous. 5answers 29k views ### Why does 1/x diverge? So for the formula \dfrac {1}{x}, If you were to add up all y values from x=1 to x=∞, wouldn't the sum approach a number because even though you are always adding, aren't you just adding ... 1answer 442 views ### Definite Dilogarithm integral \int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx Prove the following$$\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx = -3\zeta(5)+\pi^2 \frac{\zeta(3)}{3}$$where$$\operatorname{Li}^2_2(x) =\left(\int^x_0 \frac{\log(1-t)}{t}\,dt \right)^2$$2answers 508 views ### How to evaluate this integral? How would I prove that$$\int_{0}^\infty \frac{\cos (3x)}{x^2+4}dx= \frac{\pi}{4e^6}$$I changed it to$$\int_{0}^\infty \frac{\cos (3z)}{(z+2i)(z-2i)}dz$$, and so the two singularities are 2i and -... 5answers 2k views ### Question about Riemann integral and total variation Let g be Riemann integrable on [a,b], f(x)=\int_a^x g(t)dt for x \in[a,b]. Can I show that the total variation of f is equal to \int_a^b |g(x)| dx ? 3answers 1k views ### The equivalence between Cauchy integral and Riemann integral for bounded functions Definitions Suppose P\colon a=x_0<x_1<\dotsb<x_n=b is a partition of [a,b]. Let \Delta x_k=x_k-x_{k-1} and \lVert P\rVert denotes \max_{0<k\le n}\Delta x_k. The Cauchy integral ... 2answers 544 views ### If \int_0^\infty {e^{-\lambda t}f(t){\rm d}t} = 0 for all \lambda >0 then f=0 a.e.? I am studying a paper in which the author uses something like that: Let f be a bounded and Lebesgue measurable function. If$$\int_0^\infty {e^{-\lambda t}f(t)\,{\rm d}t} = 0 \qquad\text{for all} ... 1answer 15k views ### Understanding the relationship between differentiation and integration I am trying to understand the relationship between differentiation and integration. Differentiation has been introduced to me by this diagram: Which displays that the derivative of a point$x$on ... 4answers 2k views ### Integral:$ \int^\infty_{-\infty}\frac{\ln(x^{2}+1)}{x^{2}+1}dx $How to evaluate: $$\int^\infty_{-\infty}\frac{\ln(x^{2}+1)}{x^{2}+1}dx$$ Maybe we can evaluate it using the well-known result:$\int_{0}^{\frac{\pi}{2}} \ln{\sin t} \text{d}t=\int_{0}^{\frac{\pi}{2}...
What are some concrete families $\mathcal F$ of real functions that are closed under integration in the sense that for every $f \in \mathcal F$ there is $F \in \mathcal F$ such that $F'=f$? Here are ...