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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6
votes
2answers
122 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
84 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 $[0,1]^2$.
0
votes
1answer
42 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
31 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!
-1
votes
1answer
35 views

Inverse integration of a special definite integral

Hi I am facing a problem with this problem. please help.
2
votes
1answer
32 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
29 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
27 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
vote
1answer
52 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
44 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
38 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
48 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
vote
0answers
29 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
92 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 ...
0
votes
1answer
18 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
31 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
125 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
votes
0answers
45 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
votes
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 )
0
votes
0answers
22 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
98 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
113 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
70 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
54 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
43 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
69 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
74 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
69 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: $$\psi (x) = \sum_{p^m \le x}\log p \space \space \space , \space \space \space p \in \mathbb P$$ ...
6
votes
4answers
692 views

Formula for computing integrals

For computing derivative of a function, we can use the definition of a derivative, i.e. $$\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}.$$ Is there some for computing integrals too?
0
votes
1answer
35 views

Calculate a triple integral - variable changed into spherical coordinates

The problem is to calculate $$\iiint_D x^2\,dx\,dy\,dz$$ where $D$ is determined by $x^2+2y^2+z^2\le2$. solution my attempt: why can I not do it like that? I change variables, calculate the ...
3
votes
2answers
51 views

Double integral help

I'm having difficulty with a question. It says By putting $x=r\cos(\theta), y=r\sin(\theta)$, prove that $$\int_0^{\infty}\int_0^{\infty}e^{-(x^2 + 2xy\cos(\alpha)+y^2)}dx\ ...
1
vote
2answers
140 views

Real analysis question involving inhomogenous linear ODE

So I had another problem like this but the ODE was homogenous, now there is a non zero right side. I completed part (i), $\large c(x) = \int \frac{b(x)}{g(x)} dx$. I am stuck on (v). (1) is the ...
0
votes
1answer
44 views

Have I done something wrong in this integral?

I have showed most of my steps here so I hope that this is easy to follow. I have the integral $$A = C\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}Y^*(\theta, \phi)f(\theta,\phi)sin(\theta) d\theta ...
6
votes
1answer
132 views

How to prove $\int^1_0\int^1_0\frac{\log(x-x^2)-\log(y-y^2)}{(x-x^2)-(y-y^2)}dxdy=7\sum_{i=1}^\infty i^{-3}$?

How do you prove that $$\int^1_0\int^1_0\frac{\log(x-x^2)-\log(y-y^2)}{(x-x^2)-(y-y^2)}dxdy=7\sum_{i=1}^\infty i^{-3}\;\;\;\left(=7 \zeta(3)\right)~?$$ p.s. Mathematica gives a pretty good ...
2
votes
2answers
49 views

Help changing the order of integration

So I need to change the order of integration. I am giving the following limits, $1 \leq x \leq 9$ and $\sqrt{x} \leq y \leq 4$. I am having no luck solving this one. Any help would be greatly ...
0
votes
3answers
48 views

How to find $\int\sqrt{(26x-x^2)}dx $

How do I find $\int \sqrt{(26x-x^2)} dx $ Is this an integration by parts question? Thanks, --Nick
1
vote
1answer
227 views

Midpoint approximation over/under estimation

So left handed approximation underestimates the area under a increasing curve and over estimates for decreasing curves. And right handed approximation overestimates for increasing curves and ...
2
votes
1answer
108 views

Question about integration on a box

Let $Q \subseteq \mathbb{R}^n$, and $f: Q \to \mathbb{R} $ is integrable over $Q$. $f \geq 0$. if $A \subseteq Q$, then $\int_Q f \geq \int_A f $ Attempt: say $\epsilon > 0$ Let $P_1$ be a ...
-1
votes
0answers
13 views

Integral Evaluation with MATLAB-Mupad (triple and lesser degree integrals)

https://www.wolframalpha.com/input/?i=integral+of+2c%28x%5E2%2By%5E2%29%28√%28a%5E2+-+x%5E2+-+y%5E2%29%29+with+respect+to+y+from+-√%28a%5E2+-+x%5E2%29++to+√%28a%5E2+-+x%5E2%29 Here is a link to the ...
1
vote
2answers
90 views

Equality of integrals: $ \int_{0}^{\infty} \frac {1}{1+x^2} \, \mathrm{d}x = 2 \cdot \int_{0}^{1} \frac {1}{1+x^2} \, \mathrm{d}x $

In Street-Fighting Mathematics (page 16), Prof. Sanjoy Mahajan states that $$ \displaystyle\int_{0}^{\infty} \frac {1}{1+x^2} \, \mathrm{d}x = 2 \cdot \displaystyle\int_{0}^{1} \frac {1}{1+x^2} \, ...
1
vote
1answer
31 views

Does it Make Sense to Use the Variable of Integration as a Bound?

I can't for the life of me seem to decide if using the variable of integration as a bound makes sense. For instance, integrating $y=x$ from $0$ to $x$. I don't think it does… But I'm not sure.
2
votes
3answers
82 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 ...
0
votes
1answer
18 views

Numerical integration: Quadrature method which one to use?

Since “it depends” is the proper answer to a question about what quadrature method to use in evaluating an integral, what are the things that one should consider when making a choice.
1
vote
1answer
39 views

The Gaussian Integral

Hi I am trying to calculate the expected value of $$ \mathbb{E}\big[x_i x_j...x_N\big]=\int_{-\infty}^\infty x_ix_jx_k...x_N \exp\bigg({-\sum_{i,j=1}^N\frac{1}{2}x^\top_i A_{ij}x_j}-\sum_{i=1}^Nh_i ...
0
votes
1answer
47 views

Expansion of 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} show that \begin{equation} I(x)= 4+ \frac{8}{\pi}x +O(x^{2}) ...
0
votes
1answer
22 views

natural log integral question dx/(13-x)

Just wondering why the answer to the integral: $$\int \frac{\mathrm{d}x}{13-x}$$ is $-\ln|x-13|$ as opposed to $-\ln|13-x|$. Why do the $13$ and $x$ get switched?
0
votes
1answer
20 views

Solving a double integral using substitution

The problem: Evaluate $$\iint_{D}(x+y)^2(x-y)^5\:\mathrm{d}x\:\mathrm{d}y,$$ where $D$ is a rectangle with vertices in $(0, 1), (1, 0), (1, 2), (2, 1)$. So I drew the square and thought up this ...
1
vote
1answer
16 views

Finding all continuous solutions to an integral

I need help with both parts of this problem. Part (i) seems obvious, because the integrand $f(t)$ would become $F(t)$, which is obviously differentiable because its derivative is $f(t)$ by ...