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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0
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1answer
20 views

Don´t know how to start proving this formula.

\begin{equation*} \int \frac{\cos ^{m}x}{\sin ^{n}x}dx=-\frac{\cos ^{m+1}x}{(n-1)\sin ^{n-1}x}- \frac{m-n+2}{n-1}\int \frac{\cos ^{m}x}{\sin ^{n-2}x}dx+C,\qquad (n\neq 1). \end{equation*} I`d like to ...
-1
votes
0answers
29 views
+50

Question concerning the integrability of a function

Let $f: [0,1]^2 \to \mathbb{R}$ be a function such that $$ f(x,y) = \left\{ \begin{array}{lr} 1 & : x \in \mathbb{Q} \\ 2y & : x \notin \mathbb{Q} \end{array} ...
7
votes
1answer
102 views

Calculate the following Integral (Please Help)

I am trying to calculate: $$\int_0^1 \frac{\ln(1-x+x^2)}{x-x^2}dx$$ I am not looking for an answer but simply a nudge in the right direction. A stradegy, just something that would get me started. ...
1
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4answers
43 views

Are discontinuous functions integrable? And integral of every continuous function continuous?

According to me answer of second part is yes as integration simply means area under curve.
2
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2answers
30 views

Integration by reduction

I have learnt how to integrate by reduction formula but this one seems to give me hell someone to lift me by telling me what to do or simply to solve it. \begin{equation} I_n=\int\sec^n x\,dx ...
2
votes
1answer
27 views

how ro prove f(x,y) is integrable in $[a,b]\times[c,d]$

If there exits a $f(x,y)$ in $\mathbb{R}^2$,and if we fix any $x$ in $[a,b]$, then $f(x,y)$ is increasing as $y$ increases. Also, if we fix any $y$ in $[c,d]$,the $f(x,y)$ is increasing as $x$ ...
0
votes
4answers
128 views

Need help with this integral $\int_0^{\pi/2}{dx\over{4+9\cos^2x}}$

I'm trying to solve this integral - $$\int_0^{\pi/2}{dx\over{4+9\cos^2x}}$$ I've started by dividing numerator and denominator by $\cos^2x$ ...
3
votes
3answers
62 views

Convergence of $\int_{0}^{+\infty}\ln(1+\frac{1}{t^2})$

Study the convergence of $\int_{0}^{+\infty}\ln(1+\frac{1}{t^2})dt$ For $+\infty$ case it's easy we have $\ln(1+\frac{1}{t^2})\sim \frac{1}{t^2}$ For $0$ case I feel it's ...
1
vote
0answers
32 views

Computing the values of $ W_\alpha(n):=\int_0^\pi x^{\alpha-1}\sin(x)^{2n} $

Define the following integral as $$ W_\alpha(n):=\int_0^\pi x^{\alpha-1}\sin(x)^{n}\,,\quad V_\alpha(n):=\int_0^\pi x^{\alpha-1}\cos(x)^{n} $$ where $n \in\mathbb{N}$. Now in the base case ...
3
votes
0answers
26 views

Fundamental Theorem of Calculus and inverse..

If $F(x)$ is defined as $$F(x)= \int_{a}^{x} f(t) dt$$ calculate $(F^{-1})'(y)$ in terms of $f$. I have been working on this for a while now, does the aanswer to this incorporate the Inverse ...
9
votes
2answers
317 views

Another math contest problem: $\int_0^{\frac{\ln^22}4}\,\frac{\arccos\frac{\exp\sqrt x}{\sqrt2}}{1-\exp\sqrt{4\,x}}dx$

Prove: $$ {\Large\int_{0}^{\ln^{2}\left(2\right) \over4}}\, \frac{\arccos\left(\vphantom{\huge A} {\exp\left(\vphantom{\large A}\sqrt{x\,}\right) \over \sqrt{\vphantom{\large A}2\,}}\right)} ...
0
votes
1answer
13 views

Question about Riemann Integration and the indicator function

Let $S \subseteq \mathbb{R}^n$. Suppose $\chi_S$ is integrable and $\int_Q \chi_S = 1 $ for some rectangle $Q$ such that $S \subseteq Q $. Let $\epsilon > 0 $ be given, I want to ask how can I ...
0
votes
2answers
653 views

Find the surface area of the part of the paraboloid $ z=5-(x^2 + y^2)$ that lies between the planes $z=0$ and $z=1$.

I have the following math question: Find the surface area of the part of the paraboloid $ z=5-(x^2 + y^2)$ that lies between the planes $z=0$ and $z=1$. So far i have computed $\sqrt{fx^2+fy^2+1}$ ...
9
votes
4answers
254 views

How to prove $f\equiv 0$ without Weierstrass theorem?

Let $\,f:[0,1] \to \mathbb{R}$ continuous. Show that: If $$\int_0 ^1 x^k f(x)\, dx=0,$$ for all $k\in\mathbb N$, then $f\equiv 0$. I know we can use Weierstrass theorem but I'd like to ...
0
votes
1answer
39 views

How to prove that $\max\{f,g\}$ is Riemann integrable? [duplicate]

If f(x) and g(x) are Riemann integrable in [a,b], why $h(x)=\max\{f(x),g(x)\}$ is still Riemann integrable in [a,b]? Or maybe it is wrong?
2
votes
1answer
97 views

Another way to prove that $\sqrt{f}$ is Riemann integrable

[This is an exercise of Measure and Integration. I am repeating this in my vacation.] Define $f:[a,b] \rightarrow R$ a function such that $f(x) \geq 0$ over $[a,b]$ and f is R-Integrable in [a,b]. ...
6
votes
2answers
114 views

Compute $I=\int_0^{+\infty}\frac{\arctan(t)}{e^{\pi t}-1}dt$

I would like to compute $\displaystyle I=\int_0^{+\infty}\frac{\arctan(t)}{e^{\pi t}-1}dt$ Let $D=(0,+\infty)$, I have $\frac{1}{e^{-\pi t}-1}=\frac{e^{-\pi t}}{1-e^{-\pi t}}$ So ...
1
vote
1answer
83 views

munkres analysis integration question

Let $[0,1]^2 = [0,1] \times [0,1]$. Let $f: [0,1]^2 \to \mathbb{R}$ be defined by setting $f(x,y)=0$ if $y \neq x$, and $f(x,y) = 1$ if $y=x$. Show that $f$ is integrable over $[0,1]^2$.
2
votes
1answer
48 views

Integration by expansion

Consider the integral \begin{equation} I(x)= \frac{1}{\pi} \int^{\pi}_{0} \sin(x\sin t) \,dt \end{equation} show that \begin{equation} I(x)= \frac{2x}{\pi} +O(x^{3}) \end{equation} as ...
-1
votes
1answer
33 views

Laplace's Method (Integration)

Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt \end{equation} Use Laplace's Method to show that \begin{equation} I(x) \sim ...
0
votes
1answer
30 views

a question about integral? I have no idea about that!

If f(x) and g(x) are integrable in [a,b], can we say that f(x)g(x) is still integrable in [a,b]? I am referring to Riemann integration!
2
votes
2answers
103 views

Why do we bother with $u$-substitution?

This question has bothered me ever since I learned $u$-substitution (A note here: I have no formal education at this level, so I may definitely have missed something). The method is presented as an ...
-1
votes
1answer
34 views

Inverse integration of a special definite integral

Hi I am facing a problem with this problem. please help.
2
votes
1answer
27 views

Re-interpreting double integral as a Type II Region $\mathrm{d}y\,\mathrm{d}x$ vs $\mathrm{d}x\,\mathrm{d}y$

I have the following Double Integral:$\iint_Dx\cos y\space\mathrm{d}A$ where $a$ is bounded by $x=1,y=0,y=x^2$. Interpreting this region as a Type one region, it is easy to conclude $R=\{(x,y)\mid ...
2
votes
1answer
27 views

Using Polar Integrals to find Volume of surface

Here's the Question and the work that I've done so far to solve it: Use polar coordinates to find the volume of the given solid. Enclosed by the hyperboloid $ −x^2 − y^2 + z^2 = 61$ and the plane $z ...
0
votes
2answers
23 views

How to integrate by reduction method

How to evaluate the integrals of (a) $(\ln(x))^n$ (b) $x^ne^{ax}$ where $a$ is a constant By reduction formula
1
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1answer
43 views

Suppose $f(x)\in L_1$ - Prove that $\lim_{n\rightarrow\infty}\int_0^\infty f(x)\cos(nx)dx = 0$

Assuming knowledge of the cyclic behavior of $cos(x)$, integration by parts, and $\int_0^{\infty} f<\infty$ is enough here? Consider \begin{align} & \int_0^\infty f(x)\cos(nx)dx = ...
0
votes
3answers
40 views

How to solve integration of $\int x(x^2+k^2)^{-1/2} \, dx$?

As said in title, how do you solve integral $\int x(x^2+k^2)^{-1/2}\,dx$ where $k$ is some constant?
1
vote
1answer
35 views

How do the steps of this definite integral work?

Sorry if this is a really basic question but I can't seem to get my head around the steps involved in this integration at all. My equation to be integrated is as follows: ${ds \over s}=\mu dt$ ...
4
votes
1answer
234 views

Confusion related to integral of a Gaussian

I am a bit confused about calculating the integral of a Gaussian $$\int_{-\infty}^{\infty}e^{-x^{2}+bx+c}\:dx=\sqrt{\pi}e^{\frac{b^{2}}{4}+c}$$ Given above is the integral of a Gaussian. The ...
0
votes
1answer
43 views

Are there integrals you can't solve without inverse hyperbolic substitution?

Are there any integrals that can't be solved with only trig substitution? An integral that requires you to use a hyperbolic or inverse hyperbolic substitution?
1
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0answers
24 views

Find the power series for a definite integral

I am a bit unsure when integration is used together with summation. Here is my question: Find power series for $\int_0^{1} \frac{\sin x}{x}dx$ in the form $\sum_{k=1}^{\infty} a_kx^k$ Here is what I ...
1
vote
2answers
84 views

Evaluate $\iiint xyz$

Evaluate $$\iiint_E xyz\, dV$$ where $E$ is the solid: $0\leq z\leq 9,\,0\leq y\leq z,\, 0\leq x \leq y.$ I am having a hard time drawing a picture of this solid $E$ to find out what the ...
2
votes
0answers
25 views

The Fourier transform of a power of the absolute value function (and a related integral)

What (Fourier-analytic?) methods would I use to compute the following two integrals? $\displaystyle\int_{\mathbb{R}} e^{2 \pi i t} |t|^a dt \:\:\:\:\:\:\: \:\:\:\:\:\:\: \text{ and } ...
0
votes
1answer
15 views

Parametrize plane and get surface area

Find a parametrization of the surface: $y + 2z = 2$ inside the cylinder $x^2 + y^2 = 1$. Then, compute its surface area. I'm having trouble finding the parametrization of the surface. I don't think ...
1
vote
0answers
30 views

Integral of $\sin^{-1}(1/2 - \sin x) dx$

Obviously the solution exists for the integral $\int\sin^{-1}(\sin x) dx$, but does the solution exist for $\int\sin^{-1}(1/2 - \sin x) dx$? Or, for that matter, $\int\sin^{-1}(\alpha - \sin x) dx$ ...
2
votes
1answer
121 views

LogSine Integral $I=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) d\theta$

These are known as LogSine integrals at $2\pi/3$, so I will call the integral Ls as this is common in the literature. I am trying to prove $$ Ls=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) ...
2
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0answers
28 views

Log Cosine Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$

$$ I=\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3). $$ Note $\zeta(3)$ is given by $$ \zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I have a ...
0
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0answers
17 views

Construction antiderivative analytical

Find the exact area between the curves $y=x^2$ and $y=2-x^2$ with antiderivative analytical method. (fundamental theorem of calculus )
3
votes
1answer
28 views

Tricky looking integration (after separation of variables)?

I've come across something in my notes that jumps from: $${d\rho \over dz} = \sqrt{\left({\rho \over C}\right)^2 - 1}$$ to: $$\rho(z) = C \cosh\frac{z-z_0}{C}$$ I know that separation of variables ...
0
votes
0answers
18 views

Showing equivalence of weak convergence on closed and open intervals

Quick question. Let $I$ be an open bounded subset of $\mathbb{R}^{n}$. If I am given that $u_{m},u \in W^{1,\infty}(I)$ and I want to show that $u_{m} \rightharpoonup^{*} u$ in $L^{\infty}(I)$. Then I ...
4
votes
3answers
94 views

Limit of the integral of $\frac{x^n+1}{x^n+2}$

Consider the following integral: \begin{align} F(x) = \lim_{n\rightarrow\infty}\int f_n(x)dx = \lim_{n\rightarrow\infty}\int\frac{x^n+1}{x^n+2}dx \end{align} How does one evaluate this integral ...
1
vote
2answers
109 views

Integrate $\int^{ln(2)}_0 (3e^u - e^{2u} - 2)\sin(nu)du$

I'm having trouble integrating this function $$\begin{equation} \begin{split} f(x) & = \int^1_0x(1-x)\sqrt{1+x}\sqrt{1+x}\sin(n \ln(1+x))/[(1+x)^2] = \\ & = ...
5
votes
1answer
59 views

Problem with a sequence with multiple integrals [duplicate]

How to compute the following limit, $\displaystyle \lim\limits_{n \to \infty} \int_0^1 \int_0^1 \ldots \int_0^1 \sin \bigg(\frac{x_1+x_2+\ldots+x_n}{n}\bigg)\,dx_1 \,dx_2 \ldots \,dx_n$ ? I will ...
3
votes
0answers
47 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...
1
vote
1answer
39 views

Compute $\int_{0}^x \vert \sin(t)\vert dt$ for $x\in \mathbb{R^+}$

Let $x\in \mathbb{R^+}$, compute $$\int_{0}^x \vert \sin(t)\vert dt$$ I tried like this : $$ \int_{0}^x \vert \sin(t)\vert dt=\int_0^{\lfloor \frac{x}{\pi}\rfloor \pi}\vert \sin(t)\vert ...
2
votes
1answer
65 views

Hypergeometric Function simple identity

I must proove this property but I really have no idea of how to proove it: $${}_2F_1(a,b;c;z)=(1-z)^{-a}{}_2F_1(a,c-b;c,\frac{-z}{1-z}) $$ It seems its a 'simple' property, but I haven't been able to ...
1
vote
2answers
70 views

The Cantor set and integrability of $\frac{1}{x}$

Let $\chi_C$ be the characteristic function of the standard Cantor set fully contained in the interval $[0,1]$. The problem is to resolve if $\lim\limits_{\varepsilon\to 0^{+}} ...
1
vote
2answers
118 views

Using implicit function theorem without using the inverse function theorem.

Let $f:U\rightarrow \mathbb{R}$ defined on the open set $U \subset \mathbb{R}^m$. It the function $g(x):U\rightarrow \mathbb{R}$, given by the expression $$g(x)= \int_{0}^{f(x)} (t^2 + 1)dt,$$ of ...
2
votes
1answer
43 views

Question on the Prime Number Theorem (the Tchebychev Function) [duplicate]

This has been giving me nothing but a headache: Let the Tchebychev Function, $\psi (x)$ be defined: $$\psi (x) = \sum_{p^m \le x}\log p \space \space \space , \space \space \space p \in \mathbb P$$ ...