Tagged Questions

All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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1answer
81 views

How to solve this definite integral?

$$\int _{-\pi }^{\pi }\left(\frac{x^3+\cos \left(x\right)+1}{1+\cos \left(x\right)}\right)\,\mathrm{d}x$$ Any calculation program can't do it.
15
votes
8answers
4k views

Why do we require radians in calculus?

I think this is just something I've grown used to but can't remember any proof. When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does ...
1
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1answer
68 views

Using integration and polar coordinates to find the volume of a torus

How would I find the volume of the body formed by revolving the circle $r = f(\theta) = \cos\theta$ about the line $\theta = \frac{\pi}{2}$ ? (This is the circle of radius $1$ centered at $(0,1)$ ...
1
vote
1answer
61 views

Double integral, calculate area of surface.

Calculate the conical surface: $$ 3z^2 = x^2 + y^2,\quad\quad (0 ≤ z ≤2) $$ I understand that to calculate ds i need to do partial integration as so: $$ 6z{\partial z\over\partial x}=2x,\quad\quad ...
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3answers
119 views

Integrating $\int _0^\pi \frac{1}{1+\sin^2(\theta)}$ using Cauchy's formula

I need to evaluate $\displaystyle \int _0^\pi \frac{d\theta}{1+\sin^2(\theta)}$ by using Cauchy's integral formula, and the substitution $z = e^{i\theta}.$ So far, I have that $$d\theta = ...
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3answers
95 views

Evaluate the integral. $\int x^2 \log(4x) dx$

The problem is $\int x^2 \log(4x) dx$ Here $\ln$ refers to the natural logarithm. So far, I know $u = x^2$ and $du = 2x (dx)$. So $dv = \ln(4x) dx$ and $v = 1/x$, but I don't know where to go from ...
0
votes
2answers
59 views

Area between the curves $x=(5/3)y$ and $x=\sqrt{1+y^2}$

$x=(5/3)y$ $x=\sqrt{1+y^2}$ Set up and evaluate an integral expression with respect to y that give the area of S. S is the area between these 2 curves and the axes. I have already found the point ...
2
votes
2answers
803 views

Definite integral by u-substitution (1/u^2), u given

$$\int_{-3}^0 \frac{-8x}{(2x^2+3)^2}dx; u=2x^2+3$$ I need help solving this integral -- I'm completely bewildered. I've attempted it many times already and I don't know what I am doing wrong in my ...
2
votes
1answer
59 views

Minimizing $\int_0^1\left\lvert -x + e^\varphi\right\rvert d\varphi$

Which value of $x$ minimizes the following integral? $$\int_0^1\left\lvert -x + e^\varphi\right\rvert d\varphi$$ Using computational mathematics software I've seen the value of $x$ should be in ...
1
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1answer
102 views

Difficult Integrand

I am trying to integrating the following function. I am getting nowhere. $$2^{-n} n \int_0^1 w \log ^2(w) \left(2 w+w \log ^2(w)-2 w \log (w)\right)^{n-1} dw$$, Any Suggestions? Thanks for your ...
5
votes
2answers
96 views

is there any history at all for this notation of partial anti-derivatives?

i have searched but can not find examples of any published book or online articles that use this notation: $$\int f(x,y) \partial x$$ seems it would be useful for example here: $$\int_I\int_J ...
0
votes
2answers
47 views

How to calculate $\int udF(u)$?

According to a geostatistics book, \begin{array}{} Q(z) &= \int_{z}^{\infty} u dF(u)\\ & = \bigg[-uT(u)\bigg]_{z}^{\infty} + \int_{z}^{\infty} T(u)du \end{array} where $F(u)$ is a ...
2
votes
2answers
65 views

Practical motivation for analysis at school

I want to show pupils at an age of 14 to 17 and who do not like math what they can use analysis/analytic geometry in a job. I could not remember an impressive example for most topics. Let's assume ...
1
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2answers
48 views

Volume of a “tent”

In $(x,y,z)$-space, the ground is the $(x,y)$-plane $z=0$. Above the ground is constructed a giant tent whose height over $(x,y)$ is $$ h(x,y)=z=\frac{100}{1+(x^2+4y^2)^2} $$ Find the volume enclosed ...
3
votes
1answer
48 views

Integral with scalar product question

Quick question about inner products and integration: Assume that we have a function $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ where $\Omega \subset ...
0
votes
2answers
63 views

Double Integral - Sketch region and evaluate

Sketch the region of integration and evaluate the integral: $$\int_1^2 \int_y^{y^2} dx \, dy$$ I understand how to take the integral, but the region of integration seems like it has no bounds. Like ...
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0answers
56 views

I need to integrate with $\delta$ against something that isn't a test function!

In relation to ``How does integration over $\delta^{(n)}(x)$ work?,'' I need to evaluate $\int_{-a}^{a}f(x)\delta^{(n)}(x)\, dx$. However, while my $f$ is smooth on its domain, it can't be a test ...
0
votes
1answer
34 views

Convolution of fraction function

I know that convolution is defined: $$f*g=\int f(x-y)\cdot g(y) \, dy $$ How to develop below functions to convolution equation $$\int {f(x-y) \over g(y)} \, dy =\text{ ???}$$ and $$\int {f(x-y) ...
1
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2answers
43 views

Volume of a solid in $xyz$-space

Calculate the volume of the solid in $xyz$-space bounded by the surfaces $$ \displaystyle z=\frac{1}{x^2+y^2+1} and\:z=\frac{1}{x^2+y^2+4} $$ I haven't done triple integrals for a long time. But it ...
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0answers
42 views

Integration by parts on a sum of a product of partial derivatives

Let $\Omega$ be a region $\subset \subset \mathbb{R}^{n}$, and suppose that both $u_{\infty}$ and $\varphi$ $\in C_{0}^{\infty}(\Omega)$. Then, I want to perform integration by parts on the ...
2
votes
2answers
65 views

How could this Integration be solved? Any possible Steps.

$$ \int \frac{\cos x + x\cdot \sin x}{x\cdot (x + \cos x)}dx $$ Any steps leading to a simple answer would be really appreciated. Thanks !
2
votes
1answer
67 views

The antiderivative of $\sin(1/x)$

How to prove that the function $f(x)=\sin\frac{1}{x}$ for $x\neq 0,f(0)=0$ has an antiderivative? This means $F(x)=\int^{x}_{0}\sin(1/t)dt$ has derivative $0$ at $x=0$, but I have no idea how to prove ...
-2
votes
2answers
69 views

Some Integrations Hardly +2 Level

I am a programmer and definitely weak at Integration. Can some one help me out with these integrations? It would be a real help! Thanks :) I don't simply want the answers but maybe a few steps on how ...
3
votes
1answer
115 views

Evaluating trigonometric integral and Cauchy's Theorem

I am trying to evaluate the following integral: $\int_0 ^\pi {d\theta\over{1+\sin^2\theta}}$ I tried using the substitution of $\sin\theta={1\over 2i}(z-1/z)$, where $z=e^{i\theta}$, and ...
5
votes
6answers
381 views

How to integrate $\displaystyle 1-e^{-1/x^2}$?

How to integrate $\displaystyle 1-e^{-1/x^2}$ ? as hint is given: $\displaystyle\int_{\mathbb R}e^{-x^2/2}=\sqrt{2\pi}$ If i substitute $u=\dfrac{1}{x}$, it doesn't bring anything: ...
1
vote
1answer
53 views

Continuity of integral of continuous functions

Let $f\in L^1(\mathbb{R})$. Show that the function $g$ defined on $\mathbb{R}$ by $$ g(x) = \int_{\mathbb{R}} \sin(xy)f(y)dy$$ is well defined and continuous on the real line. So I want to prove ...
1
vote
3answers
111 views

Help with integral $\int\frac{1}{\sqrt{\tan x}}dx$

I tried to solve by parts but it did not help.
2
votes
1answer
110 views

Show Riemann integrable of composite function with possible discontinuity

Let $f$ be nonnegative continuous function on $[a,b]$ and $g$ be a convex function on $[0,\infty)$. Prove $g\circ f\in R[a,b]$. If $g$ is continuous on $[0,\infty)$, then it is trivial. Now the ...
1
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2answers
63 views

Help with integral with $\arcsin x$.

$$\int \frac{(1+x^2)\arcsin x}{x^2\sqrt{1-x^2}}dx$$ I saw that $$(\arcsin x)'=\frac{1}{\sqrt{1-x^2}}$$ and I tried to solve it "by parts"
0
votes
2answers
62 views

Integral of $\int \frac{\sqrt{{1+x²}}}{2 + x²} dx$? [closed]

How can i find the following integral : $$\int \frac{\sqrt{{1+x²}}}{2 + x²} dx$$
1
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0answers
22 views

Evaluation of an Integral in Vector Analysis

I'm trying to calculate an individual probability $P(\hat{a})$ from a joint probability $P(\hat{a},\hat{b})$ in a physics application, where $\hat{a},\hat{b}$ are unit vectors. I need to evaluate the ...
0
votes
3answers
112 views

Evaluating $\int \frac{\operatorname d \! x}{\sin^4{x}+\cos^4{x}+\sin^2{x}\cos^2{x}}$

How do you integrate $$\frac{1}{\sin^4{x}+\cos^4{x}+\sin^2{x}\cos^2{x}}$$ or simply $$\frac{1}{1-\left(\frac{\sin{2x}}{2}\right)^2}.$$
2
votes
1answer
107 views

Swapping integrals and limits

Let $f(x)$ such that $$f(tx)e^{-x^2}dx$$ is integrable on $[-\infty, \infty]$ $\forall t>0$. Also, $f(x)$ is continuous around 0. I'm trying to prove that $$\lim_{t\to0^+} ...
3
votes
2answers
57 views

Integrate the following function:

Evaluate: $$\int \frac{1}{ \cos^4x+ \sin^4x}dx$$ Tried making numerator $\sin^2x+\cos^2x$ making numerator $(\sin^2x+\cos^2x)^2-2\sin^2x\cos^2x$ Dividing throughout by $cos^4x$ Thank you in ...
1
vote
2answers
52 views

Can an integral be proved to have a finite value if an upper bound of the integrand has a finite value for improper integrals?

Can we say $ \int_{0}^{\infty} f(x) \text{dx} < \infty$ if $\exists \quad g(x) : \quad g(x)\geq f(x)\; \forall x \in \mathbb{R}$ and $ \int_{0}^{\infty} g(x) \text{dx}$ is finite. If yes, ...
3
votes
3answers
121 views

Show $f$ in Riemann integrable

Let $\displaystyle f(x)=\begin{cases} \frac{1}{n}, & \text{if }x=\frac{m}{n},m,n\in\mathbb{N}\text{ and m and n has no common divisor} \\ 0, & \text{otherwise} \end{cases}$ Show $f\in ...
1
vote
1answer
38 views

Upper incomplete gamma integral

I would like to know whether the following relation is correct or not? $\frac{d}{dz}\Gamma(w,\mu z)= -\mu^wz^{w-1}e^{-\mu z}$, where $\Gamma(w,\mu z)$ is the upper incomplete gamma integral. Can ...
0
votes
0answers
27 views

Complex number contour integral

Determine the contour integral ∫Ѱ 1/z dx, where Ѱ is the positively oriented unit circle with centre at -2 given that Ѱ(t) = -2 + e^(it), 0<=t<=2pii. I understand that Ѱ is the positively ...
3
votes
1answer
66 views

How does integration over $\delta^{(n)}(x)$ work?

For a math paper I need to be able to evaluate $\int_{-a}^{a}\delta^{(n)}(x)\ f(x)\ dx$ for differentiable $f$. I know that it is 'supposed' to equal $(-1)^nf^{(n)}(0)$: $$\int_{-a}^a\delta^{(n)}(x)\ ...
1
vote
2answers
38 views

Definition of Lebesgue integrability

As I understand it, a function $f : \mathbb{R} \to [-\infty, \infty]$, (or more generally from any measure space $X$, is integrable (Lebesgue integrable) if $\int f d\mu$ is finite. Why is it ...
1
vote
2answers
129 views

Is every Riemann-integrable function lebesgue integrable?

I thought every Riemann integrable function is lebesgue integrable. But here, a user comments that the statement is not correct. I'm confused. Can someone clarify?
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votes
1answer
70 views

Pulling out variables through the integral sign sometimes it's okay sometimes it's not okay

clearly a bad idea to do this.. $$\int 4yx^3dx = yx^4$$ ..plus possibly a constant or a function of y... specially bad, for example, if $y$ happened to be $y=x^3$ or any non constant function of $x$ ...
0
votes
1answer
656 views

How to calculate the polar arc length of the entire cardioid $r=a(1-\cos\theta)$

I'm having a bit of an issue calculating the arc length of $r = a(1-\cos\theta)$. I'll begin by listing the steps I made in my attempt to solve this exercise. We know that the arc length formula ...
2
votes
1answer
66 views

Magical test for convergence of improper integrals?

I found this article while surfing the web. I hope it's not some kind of joke, because if it is it really fooled me. I'm trying to figure out the proof of theorem 2.3 I don't understand how the ...
1
vote
5answers
41 views

How to compute the integral $(1+t^2) / (1-t^2) $

compute $\displaystyle\int \frac{1+t^2}{1-t^2}dt$ I tried splittting them to two parts, and computed $\displaystyle\int \frac{1}{1-t^2}dt$ using trig sub, but i don't know how to compute the second ...
1
vote
1answer
55 views

Volume of $\cos(x)$ around y-axis using numerical integration

Okay guys, so I have a question about numerical integration relating to the equation $y=\cos(x)$. The general equation is as follows: $$\pi \int_0^{2 \pi } \left[\cos^{-1}(y)\right]^2 \, dy$$The ...
1
vote
0answers
24 views

Hellinger Integral properties

Let $\mu , \nu$ be two probability measures on $(\Omega , \mathcal{F}).$ Suppose we have a probability measure $\lambda$ such that both $\mu , \nu$ are absolutely continuous with respect to ...
2
votes
1answer
40 views

Absolutely Continuous measures and Hellinger integral

Let $\mu , \nu$ be two probability measures on $(\Omega , \mathcal{F}).$ Suppose we have a probability measure $\lambda$ such that both $\mu , \nu$ are absolutely continuous with respect to ...
2
votes
1answer
63 views

Evalutating the following integral: $\int\frac{\ln(x) dx}{x+4x\ln^2(x)}$

I want to find this integral but I work on it an hour now still no clue... please just give me a HINT and not the answer so that I can find it on my own and learn via this process. Here is the ...
2
votes
1answer
82 views

Evaluating an integral with Laplace

We need to evaluate the following integral: $$\int_{0}^{\infty}\frac{\cos(tx)}{x^2+a^2}dx$$ There is the following note: "You may interchange taking the Laplace transform and integrating." I have ...