Questions tagged [integration]
For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.
76,934 questions
Score of 1
1 answer
119 views
Binding $\int_{1/2}^{1} f(x) dx$, if $f'(x)=\frac{192x^3}{2+\sin^4(\pi x)}$ and $f(1/2)=0$
IIT JEE (2015) asked one to find $m$ and $M$ if
$$m<\int_{1/2}^1 f(x) dx<M,\quad f(1/2)=0 \tag{1},$$
where $$f'(x)=\frac{192x^3}{2+\sin^4(\pi x)}\tag{2}$$
the options are $$(a): m=13, M=24;~ (b):...
- reputation score 50646
Score of 1
1 answer
45 views
For which function class are differentiation and integration inverses in the Henstock-Kurzweil case?
Suppose that
$$\mathcal A \subseteq \operatorname{Hom}(\mathbb R, \mathbb R)$$
and that for every $a, x \in \mathbb R$,
$$
\frac{\mathrm d}{\mathrm dx}\int_a^x f(t) \; \mathrm dt = f(x),
\qquad
\...
- reputation score 3966
Score of 1
1 answer
188 views
An integral-based geometric proof of the sum of squares
My initial construction
For each integer $k \in \{1, 2, \ldots, n\}$, consider the rectangle
$$R_k = [k-1, k] \times [0, k^2].$$
Each rectangle is treated independently. The area of $R_k$ is $k^2$.
...
- reputation score 178
Score of 4
1 answer
261 views
how to evaluate integral $\int_0^\pi\left(\tan^{-1}(1+a\cos(x))\right)^b\mathrm{d}x$?
Concerning integral
$$I=\int_0^\pi\left(\tan^{-1}(1+a\cos(x))\right)^b\mathrm{d}x,$$
where $a\in\mathbb{R}, b\in \mathbb{Z}^+$ are parameters. This is the generalization of a Cambridge integration ...
- reputation score 1502
Score of 2
0 answers
108 views
+100
Divergence of knot energy integral
I'm working with this length normalized O'hara energy in $S^3$ with Intergal
$$\frac{1}{L^2}\iint_{\gamma \times \gamma} \left(\frac{1}{d_{S^3}(\gamma(x) , \gamma(y))^2} - \frac{1}{d_{\gamma}(x,y)^2}\...
- reputation score 23
Score of 6
7 answers
266 views
$\int_{0}^{5}\frac{\sqrt{5+x}-\sqrt{5-x}}{\sqrt{5+x}+\sqrt{5-x}}\cdot\frac{dx}{\sqrt{25-x^2}}$?
I am trying to evaluate the following definite integral:
$$
I=\int_{0}^{5}
\frac{\sqrt{5+x}-\sqrt{5-x}}
{\sqrt{5+x}+\sqrt{5-x}}
\cdot\frac{dx}{\sqrt{25-x^2}}.
$$
My only progress so far is to multiply ...
- reputation score 364
Score of 1
2 answers
262 views
Repeated Partial Differentiation
I was going through a worksheet and got presented with the following problem:
Given
$$
\int_0^\infty e^{-tx}\sin(x)\,\mathrm dx = \frac{1}{1+t^2},
$$
find the resulting equation after taking the $n$...
- reputation score 31
Score of 7
0 answers
193 views
+150
Evaluation of an improper integral involving hyperbolic and trigonometric functions
I am interested in evaluating the following improper integral:
$$
\int_0^\infty
\frac{\sinh\left(2(\pi-\alpha)\nu\right) }{\sinh(\pi\nu)}
\left(
\frac{
\sinh(\beta\nu)
\sinh(\theta\nu)
}{
\sinh(\pi\...
- reputation score 77
Score of -5
0 answers
48 views
Proving $\int_{1/6}^1 \frac{\arccos x}{(1+2x)\sqrt{1+x}}\left(\frac{1}{\sqrt{1+3x}}+\frac{2}{\sqrt{x}}\right)dx=\frac{2}{15}\pi^2.$ [closed]
The problem
H . S . M . Coxeter proposed the evaluation of
$$I = \int_ {1/6}^{1}\frac {\arccos x} {\sqrt {1 + x}\, (1 +
2 x)}\left (\frac {1} {\sqrt {1 +
3 x}} + \frac {2} {\sqrt {x}} ...
- reputation score 3
Score of 4
5 answers
263 views
Integrating $\int_0^{\pi/2} \frac{\sec x}{\sqrt{a^4 + \tan^2x}} dx$
Suppose we have the integral $$\int_0^{\pi/2} \frac{\sec(x)}{\sqrt{a^4 + \tan^2x}} dx$$
Assuming $a$ is a constant, how do we integrate this expression within the given limits?
Trigonometric ...
- reputation score 51
Score of 2
4 answers
332 views
How do existence assumptions work when solving differential equations? Do they prove solutions work? Do they use conditional or biconditional steps?
When solving a differential equation (like $\frac{dy}{dx} + 3x = 5x$, or simplified to $\frac{dy}{dx}=2x$), how does the logical framework and process of solving it work in formal/rigorous mathematics?...
- reputation score 427
Score of 5
2 answers
195 views
Evaluating $ \int_0^1 \frac{\arctan x \ln^2(1-x^2)}{x} dx$ and its related series
I was trying to evaluate the integral,
$$I = \int_0^1 \frac{\arctan x \ln^2(1-x^2)}{x} dx$$
I reached a series,
$$ S = \sum_{n=1}^\infty \frac{H_n}{(n+1)^2} \left[ \psi\left(\frac{2n+5}{4}\right) - \...
- reputation score 5270
Score of 6
1 answer
281 views
Probability of explicit upper and lower bounds in the triangle inequality
Let $0<x < y\le z$ be the side lengths of a triangle whose vertices are uniformly distributed on a circle. Simulation shows that for $p > 1$ and $1\le q<r$,
$$
\Pr\!\left(
\left(\frac{x^p}{...
- reputation score 26060
Score of 0
2 answers
75 views
Corollary 4.11 from Bartle
Corollary 4.11 from Bartle's The Elements of Integration states that if $f: X \to \bar{\mathbb R}_+$ measurable, $\mu$ a measure on the $\sigma$-algebra $\mathbb X$ and $\lambda (E) = \int_E f \, d\mu$...
- reputation score 410
Score of 5
1 answer
101 views
Why isn't the integral of a scalar function on a manifold defined via direct Riemann sums of subregion volumes?
I am currently studying analysis on manifolds using Munkres' Analysis on Manifolds. My questions here stem purely from a rough, intuitive picture rather than a rigorous mathematical setup, but I am ...
- reputation score 411
