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 ...

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4
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3answers
115 views

How do I go about solving this?

I have tried substitution, but it is not working for me. $$ \int_0^\pi \frac{dx}{\sqrt{(n^2+1)}+\sin(x)+n\cos(x)}=\int_0^\pi \frac{n dx}{\sqrt{(n^2+1)}+n\sin(x)+\cos(x)}=2 $$
0
votes
1answer
9 views

Inequality of integrals with respect to a signed measure and to total variation measure

I am trying to solve this exercise: Let $\mu=\mu^+-\mu^-$ be the Hahn-Jordan decomposition of a finite signed measure on a measurable space $(X, A)$. Show that for any bounded measurable ...
1
vote
1answer
21 views

Find convergence domain of integral

I need to find convergence domain of $$\int_1^2 \! \frac{\ln(x-1)}{(4-x^2)^p} \, \mathrm{d}x$$ I've tried to use estimates like $\frac{\ln(x-1)}{(4-x^2)^p} < \frac{1}{(4-x^2)^p}$ and change of ...
0
votes
2answers
37 views

How to find $\int \frac {sinh(lnx)} {x}$

I've tried $\int \frac {sinh(lnx)} {2x} dx = \int \frac {e^{lnx}-e^{-ln{x}}} {2x} dx = \int \frac {x-e^{-ln{x}}} {2x} dx $
1
vote
3answers
51 views

Lebesgue Dominated Convergence Theorem example

For $x>0$ we have defined $$\Gamma(x):= \int_0^\infty t^{x-1}e^{-t}dt$$ Im trying to use Lebesgue's Dominated Convergence theorem to show $$\Gamma'(x):=lim_{h\rightarrow ...
-1
votes
1answer
52 views

Integral of $\int \frac{xe^x}{\sqrt{1+e^x}} dx$ [on hold]

I need help to solve this: $$\int \frac{xe^x}{\sqrt{1+e^x}} dx$$
2
votes
2answers
98 views

How can $\int_a^b f(x)dx $ exist if either $f(a)$ or $f(b)$ does not exist?

In class, I came across the integral: $$\int_0^1 \frac{dx }{\sqrt{1-x^2}}=\frac{\pi}{2}$$ This is easy enough to prove using a substitution or by recalling the derivative of $\arcsin x$. However, ...
0
votes
1answer
24 views

Given two functions, if one is greater on an interval, how to prove its integral is also greater?

Working with the definition of integral as in Spivak's Calculus book, I got a big struggle to prove the next statement (even though it seems like something very obvious). Please send help. ;_; Let ...
1
vote
2answers
45 views

Why do I get different results for the same integral?

The variables $ a, b, s, c $ are constants, so: $$ \int \left ( a \cos(s + cx) - b \sin(s + cx) \right ) dx = \frac{a\sin(s + cx) + b\cos(s + cx)}{c} +C $$ But if $c=0$ then: $$ \int \left ( a ...
2
votes
1answer
37 views

Solve integral weird upper bound approaching zero.

I am trying to solve an integral of the form $$p(0,t+\delta t)=\int_0^{\mu \delta t} f(x,t) dx$$ for $\delta t \rightarrow 0$ Intuitively, I would think that this integral has an upper bound which ...
2
votes
1answer
38 views

Proof this integral is equal to this sum.

I was able to determine the integral with trials and errors and we arrive with this sum, but was not able to proof it. So if anyone can proof it and also can offer a closed form. $$ \int_0^\infty ...
0
votes
2answers
39 views

$\int^{2 \pi}_0 \frac{1}{3+2 \cos t}dt$ using $\cos t = \frac{1}{2}\left(e^{it} + \frac{1}{e^{it}}\right)$ or using $u=\tan \frac{t}{2}$

Question : Compute the integral of $$\int^{2 \pi}_0 \frac{1}{3+2\cos t}dt$$ I am stucked on this problem since a good while. I think we could convert that real integral into complex integral and ...
2
votes
2answers
74 views

Good books on integrals [duplicate]

I'm a math student at the sixth semester and I've had my courses in calculus and complex analysis. I'm able to solve integrals with the usual techniques, e.g. with substitution. However, whenever I am ...
0
votes
1answer
46 views

How do I differentiate an improper integral?

I would like to differentiate a function of the type $\int_x^\infty f(x, t) dt$ with respect to $x$ ($f$ real or complex valued). Does differentiation under the integral sign apply? What are better ...
3
votes
1answer
91 views

Is there a closed form of this integral $ \int_0^\infty \sin(xe^{-x})dx\, $? [on hold]

I have tried by subsititution method and it got more complicate than before. Can anyone help me to evaluate this integral. $$ \int_0^\infty \sin(xe^{-x})dx\,. $$
0
votes
1answer
27 views

Why most of the books give Definite Integral Represents Area Under the curve

According to my understanding The definition of Definite Integral is: if $f(x)$ is a Continuous function in $[a \:\: b]$ and if $P$ is Partition of the Interval $[a \:\: b]$ Then $$ \lim _{\lVert P ...
1
vote
1answer
25 views

If $|f_n| \to 0$ and $f_n$ are integrable, is it true that $\int |f_n| \to 0$? [on hold]

If $|f_n| \to 0$ and $f_n$ are integrable, is it true that $\int |f_n| \to 0$?
0
votes
3answers
60 views

$\int^{2 \pi}_0 \frac{1}{ \sqrt{5}+\cos t}dt$, $\int^{2 \pi}_0 \frac{\cos^2t}{ 5-3\cos t}dt$ - Cauchy integral?

Compute the integrals $$\int^{2 \pi}_0 \frac{1}{ \sqrt{5}+\cos t}dt$$ and $$\int^{2 \pi}_0 \frac{\cos^2t}{ 5-3\cos t}dt$$ I am stucked on these problems since a good while. Is there someone is able ...
1
vote
1answer
35 views

Integral and Cauchy theorem

Question : Compute the integral of $$ \int^{2 \pi}_0 \frac{1}{3+2\cos t}dt. $$ Indication: take the path $\gamma: [0,2 \pi] \to \mathbb{C}$, $\gamma(t)=e^{it}$ and the integral of $$ \int_{\gamma} ...
0
votes
1answer
13 views

$\int_{\gamma} \frac{f(z)}{z^3}dz$ - Cauchy formula

Compute the integral $$\int_{\gamma} \frac{f(z)}{z^3}dz,$$ where $f(z)=az^3+bz^2+cz+d$ and $\gamma : [0, 4 \pi] \to \mathbb{C}$, $\gamma(t)=e^{it}$. So by the Cauchy formula $\int_{\gamma} ...
2
votes
1answer
34 views

Integrating variation of error function: $\int_1^2e^{-nx^2} dx$

Show that $$\lim_{n\to\infty} \int_1^2e^{-nx^2} dx = 0.$$ After much googling, I learned that I am working with a variation of the error function! Yay. I've never heard of it in my life and I ...
2
votes
2answers
55 views

Does an analytical form exist for the following integral

I have an integral $$f(n,a)=\int_0^{2\pi}\mathop{\mathrm{d}x}\frac{\cos(nx)\cos^2x}{1-a\cos^2x},$$ where $n$ is an even integer and $0<a<1$ is a real number. Does an analytical form exist for ...
2
votes
1answer
43 views

Argue that the iterated integral of a continuous function is continuous

Suppose that $f : [a, b] \times [c, d] \to\mathbb R$ is a continuous function. Let $$G(y)= \int_a^b f(x, y) \, dx$$ $$H(x)= \int_a^b f(x, y) \, dy$$ Prove that $G$ is continuous on $[c, d]$ and $H$ ...
1
vote
4answers
111 views

Show that if $f(x) > 0$ for all $x \in [a,b]$, then $\int_{a}^b f(x) dx > 0$

Assume $f$ is Riemann integrable and nonnegative over $[a,b]$. Show that if $f(x) > 0$ for all $x \in [a,b]$, then $\int_{a}^b f(x) dx > 0$. This seems very obvious to me. One thing I would ...
-1
votes
2answers
46 views

Example of a function that converges to 0 pointwise but integral is 3/2?

Give an example of a sequence of continuous functions $(f_n)$, $f_n : [0, 1] \to \mathbb{R}$ that converges to zero pointwise, and such that the integral of each function within the given domain is ...
1
vote
0answers
26 views

Closed-form of an integral involving a Jacobi theta function, $ \int_0^{\infty} \frac{\theta_4^{10}\left(e^{-\pi x}\right)}{1+x^2} dx $

Motivation The Jacobi theta function $\theta_4$ is defined by $$\displaystyle \theta_4(q)=\sum_{n \in \mathbb{Z}} (-1)^n q^{n^2} \tag{1}$$ For this question, set $q=\large e^{-\pi x}$ and $\theta_4 ...
2
votes
0answers
17 views

Prove $\sum_{k=1}^\infty k^{-p}f(kx)$ converges absolutely almost everywhere, where $p>0, f \in \mathcal{L}^1(\mathbb{R})$.

What I've done: $$ \int_\mathbb{R} \sum_{k=1}^\infty k^{-p}|f(kx)| = \sum_{k=1}^\infty \int_\mathbb{R} k^{-p}|f(kx)|dx = \sum_{k=1}^\infty k^{-p}\int_\mathbb{R} k^{-1}|f(y)|dy = ...
1
vote
1answer
39 views

The so-called error function defined as: $erf(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^{2}}dt$

The so-called error function is defined as: $$erf(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^{2}}dt$$ show that the function $y(x) = e^{x^2}erf(x)$ satisfies the differential equation: ...
0
votes
1answer
17 views

Finding streamlines

Find the streamlines, particle paths and streaklines when $$u=xe^{2t-z}, \, \, \, v=ye^{2t-z}, \, \, \, w=ze^{2t-z}$$ What is the track of the particle passing through $(1,1,0)$ at time $t = 0$? To ...
0
votes
1answer
24 views

Is $\{(\epsilon + \cos(x))^{2k}\}_{k\in\mathbb{N}}$ a family of good kernels?

Show that for any $0<\delta<\pi$, $$\lim_{k\to \infty} c_k\int_{\delta<|x|<\pi} \left(\epsilon + \cos(x)\right)^{2k} dx = 0 $$ where $\epsilon >0$ is some small number (for ...
0
votes
2answers
58 views

I need to integrate $\ln(x^2)$ but I can't seem to get it right.

I have an issue that requires me to integrate $\ln(x^2)$ and I know it's done through integration by parts, but I just can't seem to get it right. Does anyone know how it would work out?
1
vote
1answer
30 views

Estimate $\int_0^{10} f(x)g'(x)dx$ if $f(x) = x^2$ and $g$ has the following values [on hold]

I need help solving this problem... I believe it has something to do with either Riemann sums or integration by parts. Here is the problem. Estimate $\int_0^{10} f(x)g'(x)dx$ if $f(x) = x^2$ and $g$ ...
1
vote
1answer
32 views

Explain how L(g,P) = U(g,P) implies continuity of g.

First, let $g$ be bounded on $[a,b]$. Now, assume $\exists P$, a partition, such that $L(g,P)=U(g,P)$. I am told the correct answer to the question "describe $g$" is that $g$ is continuous on ...
0
votes
1answer
45 views

$\iint y^2 dxdy$ over circular region

Suppose I want to calculate $$\iint y^2 dxdy$$ over region outside $$C_1=x^2 + y^2 = ax$$ and inside $$C_2=x^2 + y^2 = 2ax$$. How can we perform this integral? I approached this problem using polar ...
3
votes
2answers
48 views

When do we have the formula $f(t)=e^{\lambda t}f(0)+\int_0^te^{\lambda (t-s)}g(s)ds$?

Let $g:\mathbb{R}\to \mathbb{R}$ be a continuous function. Consider the following integral equation $$f(t)=f(0)+\int_0^t\lambda f(s)ds+\int_0^tg(s)ds. \tag{1}$$ Since $g$ is continuous, Thus the ...
1
vote
1answer
51 views

How can I compute this integral $\int \cos^{2}\left(t\sqrt{x^{2}-1}\right)dx $

How can I compute this integral $$\int \cos^{2}\left(t\sqrt{x^{2}-1}\right)dx $$ Even when I use Maxima, it does not give result. Thank you very much.
2
votes
2answers
55 views

Suppose $\int_0^9f(t)dt=12$. Then is it true that $\int_0^3f(3x)dx = 12$

Suppose $\int_0^9f(t)dt=12$. Then is it true that $\int_0^3f(3x)dx = 12$ ? How do I go about figuring this out? I tried differentiating and using fundamental theorem of calculus but couldn't figure ...
0
votes
0answers
19 views

Convergence behaviour of Eichler integral

Consiger $g : \mathbb H \to \mathbb C$ a modular form of weight $2-k, k \in \frac{1}{2}\mathbb Z$. Let $z \in \mathbb H$ and consider the following integral: ...
0
votes
0answers
10 views

Line integral over a vector field

Evaluate ∫C < −y, x − 1 > dr where C is the closed piecewise continuous curve formed by the line segment joining the point A(− √ 2, √ 2) to the point B( √ 2, − √ 2) followed by the arch of the ...
0
votes
3answers
40 views

For what values of K, is the integral improper?

For what values of $K$ ($K > 0$), is the following integral improper? $$\int_{0}^{K}\frac{x}{x^2-2}$$ Now, I know that the function is undefined at $x=\sqrt{2}$. I also figured out that the ...
1
vote
1answer
29 views

How is the following integral rigorously meant to be understood?

Consider $\mathbb{R}^3$. Consider the following integral on the unit three sphere $$ \int_{S^3}\frac{1}{x^2}\,d^3x $$ where $x^2=x_1^2+x_2^2+x_3^2$. I have quite some working knowledge on integrals ...
0
votes
1answer
34 views

Simpson's rule is not producing better results than Riemann sums

I have to calculate RMS value $\sqrt {\int_0^T\frac 1T*f(t)^2dt} $ and I know from the maths that the Simpson's rule should provide better approximation of the definite integral than the Riemann sums. ...
0
votes
2answers
115 views

Anti-Derivative of $\ln(x^2 + 7)$ is kicking my butt, can anyone help?

I'm given $\ln(x^2 + 7)$ in a problem and to solve it I need to get the anti-derivative, but I haven't been able to properly calculate it. Could someone show me how to obtain this anti-derivative? It ...
3
votes
1answer
50 views

Antiderivative of $\arctan(-x^2)$

As I said in the title I'm trying to find an antiderivative of $$f(x)=\arctan(-x^2)$$ I am aware that e.g. WolframAlpha can find one, but I have no clue how to do it by hand. Can anyone give me a ...
0
votes
1answer
34 views

Using divergence theorem to calculate surface of sphere

I want to calculate: $$\iiint_V div (\overrightarrow F \cdot \space dV) $$ with $\overrightarrow F=x^3\hat i+ y^3 \hat j+z^3\hat k$ and Surface of sphere given as $x^2+y^2+z^2=r^2$ So, first I ...
1
vote
0answers
28 views

Newtons Law Of Cooling (And Heating)

Rule is: $D= A.e^{-kt}$, Where: $k,a$ are elements of real numbers, $D$ is the difference between the temperature of the item and the surrounding air, and t is the time in hours since the object ...
1
vote
2answers
72 views

Evaluate $\int \frac{\sqrt{x}+1}{\left(\sqrt[3]{x}-1\right)\sqrt[6]{x^5}}\,dx$

How should I approach this? $$\int \frac{\sqrt{x}+1}{\left(\sqrt[3]{x}-1\right)\sqrt[6]{x^5}}\,dx$$ I intend to continue with the substitution method, however, I find it difficult to understand what ...
1
vote
1answer
21 views

Double integration of a function of e to y squared where one integral has a variable

I'm not sure how to phrase the title, but I have a problem on my homework assignment that requires me to solve the following function. $\int_0^2 \int_{x^2}^4 xe^{y^2} dy dx$ WolframAlpha gives a ...
0
votes
0answers
8 views

Approximating the integral of a large product

I would like to approximate the following integral of a product: $$ I = \int dz\, f(z)\prod_{i=1}^n\left(1 - \rho_i(z)\right) $$ The functions $f$ and $\rho_i$ are differentiable for all $i$, ...
0
votes
1answer
25 views

Relation between $\lim_{n \to \infty}\int_{I}f_n(x)\:dx$ and $\int_{I}\lim_{n \to \infty}f(x)\:dx$ [on hold]

Relation between convergence and integration of sequence of a function. Let $f_n$ be a sequence of integrable functions defined on an closed interval with $$f_n(x) \to 0$$ on this interval ...