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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2
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1answer
16 views

What assumptions are needed to get two integrals close to each other?

I have functions $A,B,C$, where $\int_{\mathbb{R}} |A\cdot B - C| <\varepsilon$, and want to be able to say that $\int_{\mathbb{R}} A \approx \int_{\mathbb{R}} \frac{C}{B}$. What extra assumptions ...
0
votes
0answers
19 views

Volume by double integration?

Suppose that $h<a<0$. Show that the volume of the solid bounded by the cylinder $x^2+y^2=a^2$, the plane $z=0$ and the plane $z=x+h$ is $V=\pi a^2h$. I'm having a very hard time with ...
-3
votes
1answer
22 views

Integrate $\cot^2x-\frac{\cos^2x}{\tan^2x}$

Integrate $\int{\cot^2x-\frac{\cos^2x}{\tan^2x}}dx$
-3
votes
1answer
34 views

Integrate $\int^{1}_{0}{\sin^2x}$ [on hold]

What is the value of this integration ? $\int^{1}_{0}{\sin^2x}dx $
1
vote
1answer
23 views

Let $S_n:= \frac{b-a}{n}\sum_{i=1}^{n}f(t_{i,n})$. Prove: $\lim_{n\to\infty}S_n = \int_a^bf(x)\ dx$.

I will post the assignment and then my attempt at solving it. Let $a,b \in \mathbb{R}$ with $a<b$ and let $f: [a,b] \rightarrow \mathbb{R}$ be a continous function. We'll now define a sequence ...
3
votes
3answers
43 views

$\int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x$

My try, using $x = sec(u)$ substitution: $$ \int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x = \int \frac{\sqrt{sec^2(u) - 1}}{sec(u)}tan(u)sec(u) \mathrm{d}u = \int tan^2(u) \mathrm{d}u = tan(u) - u + C = ...
0
votes
0answers
15 views

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} ...
2
votes
1answer
37 views

Integral $\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt$

I am looking for a closed form expression for the logarithmic trigonometric integral $$ I_{n,p}=\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt \quad (n\geq 0, p\geq 0). $$ Closed form expression ...
5
votes
2answers
80 views

Integrate $I=\int_0^1\frac{\ln x}{x^n-1}dx$

Hi I am trying to obtain a closed form for$$ I_n=\int_0^1\frac{\ln x}{x^n-1}dx, \quad n\geq 1. $$ This integral is quite nice and generates many other known closed form results such as $$ ...
0
votes
0answers
6 views

Find a map T(D*)=D and Triple integral

D={(x,y,z)| (7x-3y-z)^2 +(-3x+7y-z)^2 +(-x-y+3z)^2<=100} D* = {(u,v,w)|u^2+v^2+w^2<=1} find map T(D*)=D express the triple integral of xy dx dy dz over D as an integral over D* and evaluate
1
vote
0answers
27 views

Approximate an integral

In a physics textbook, I came across the integral $$I(r_1,r_0)=\int_{r_0}^{r_1}\frac{1}{1-2m/r}\left[1-\frac{r_0^2(1-2m/r)}{r^2(1-2m/r_0)}\right]^{-1/2}dr$$ The author said that the integrand can be ...
1
vote
2answers
34 views

How to integrate $(x^2 - 1)/(x^2 + 1)$?

I have gone until separating $(x^2 - 1)/(x^2 + 1)$ into $x^2/(x^2 + 1)$ - $1/(x^2 + 1)$. The latter fraction I can substitute by $\tan u$, but how to solve the first fraction and how does it all come ...
0
votes
0answers
25 views

Is there any other integral of a special function that is undetermined?

Is there any other integral of a special function that is undetermined but yet the special function itself is continous? eg. the integral of the prime number counting function is undetermined By ...
1
vote
0answers
25 views

Integral $I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx$

Hi I'm trying to show that $$ I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx=\frac{5\pi^2}{96}. $$ We can try the substitution $u=(x^2+2)^{1/2}, du=x(2+x^2)^{-1/2}dx$ ...
4
votes
0answers
51 views

Calculate the following Integral (Please Help)

Hi, 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 ...
1
vote
2answers
37 views

Aside from this two practical technique to compute any integral, what else? [on hold]

Aside from this two practical technique to compute any integral, what else could called a fundamental method but not approximate method like Riemann Sum? These two method I've been referring to are ...
3
votes
1answer
52 views

Is a probability density function necessarily a $L^2$ function?

If a nonnegative continuous real valued function $f$ is integrable over $\mathbb{R}$ with $$\int_\mathbb{R} f\,\mathrm{d}x = 1,$$ does it hold true $$\int_\mathbb{R} f^2 \,\mathrm{d}x<\infty?$$ ...
0
votes
1answer
43 views

How to integrate $\int \frac{dy}{\sqrt{4y+\frac{1}{4y^2}+2C_1}}$?

How do I integrate $\int \frac{dy}{\sqrt{4y+\frac{1}{4y^2}+2C_1}}$, where $C_1$ is an arbitrary constant? Is this integral really complex (hard to integrate)? EDIT: This comes from DE: $dy/dx = ...
0
votes
1answer
10 views

Newton-cotes formulas help

I am having a hard time understanding how to use this formula. If given the following problem: Compute ∫ sin x dx using Simpson's rule with 3 points in the range 0 to Pi/2. Do I have to take the ...
1
vote
1answer
17 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 ...
1
vote
1answer
20 views

Calculating the center of mass in spherical coordinates

So normally, to calculate the center of mass you would use a triple integral. In my particular problem, I need to calculate the center of mass of an eight of a sphere where it's density is ...
1
vote
1answer
19 views

Primitive function tricks

Calculation here Questions: how did he get the idea to "split up" $r^2$ into $2r \cdot r/2$? is he doing integration by parts after the second = sign? I can't really follow the algebra here.
0
votes
1answer
24 views

How do I integrate this in terms of error function

How do I evaluate $$\dfrac{1}{\sqrt{4\pi t}}\int_0^{\infty}ye^{-\frac{(\xi-y)^2}{4t}}dy$$ in terms of $\text{erf}(x)$ ? I tried integration by parts but the integral seems to get complicated. I think ...
4
votes
0answers
51 views

Solving integral $\int\frac{\sin x}{1+x\cos x}dx$

How I can find the anti-derivative? $$\int\frac{\sin x}{1+x\cos x}dx$$
3
votes
3answers
219 views

William Lowell Putnam Integral Problem

Prove That $$ \frac{22}{7}-\pi= \int_0^1 \frac{x^4\,\left(1-x\right)^4}{1+x^2}$$
0
votes
0answers
45 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 (ii) and the rest. ...
1
vote
1answer
28 views

Prove that there exists only one function f such that…

Prove that there exists only one function $$\big[f\in C\left ( \left [ 0,1 \right ],\mathbb{R} \right )s.t. f(x)=\frac{2}{5}\int_{0}^{1}(x^{2}+t^{5})f(t)dt+sin(x)\big] $$
5
votes
1answer
56 views

How to evaluate $\int_0^ \infty e^{-x\sinh(t)-\frac{1}{2}t}~dt$?

$$ \int_0^ \infty e^{-x\sinh(t)-\frac{1}{2}t}~dt $$ I tried doing it by parts and looking for differentials but I just keep getting back to the original expression. I can't think of a clever ...
1
vote
1answer
37 views

Prove by using step functions: $\int_{-b}^{b}\sin(x)\ dx = 0$

The Assignment: Let $b > 0$. Prove by using step functions: $$\int_{-b}^{b}\sin(x)\ dx = 0$$ The claim itself is obvious, but I have no idea how to prove it with step functions. My idea was ...
2
votes
1answer
22 views

Convergence Question:

This is related to the Dirichlet eta function. Does $$\int_1^\infty \frac{dx}{x^z}$$ converge for $Re(z)>1$? Just wondering. If so, then does $$\int_1^2 \frac{dx}{x^z}+\int_3^4 ...
0
votes
0answers
16 views

Finding surface integral of the paraboloid and disk

Let S be the surface consisting of the paraboloid $y=x^2 + z^2$ with $0 \leq y \leq 1$, and the disk $x^2 + y^2 \leq 1$. Let $S$ have an outward orientation. Compute the double integral of $\langle ...
0
votes
1answer
41 views

integral $I=\int_{-\infty}^\infty e^{-\alpha x^{2k}}dx$

$$ I=\int_{-\infty}^\infty e^{-\alpha x^{2k}} dx $$ The last problem was ill posed, and is answered in the post! You can disregard this post!
2
votes
0answers
40 views

Integral $\int_0^{\pi/3}\log\bigg( \frac{1+2\cos\theta}{2}+\sqrt{\left( \frac{1+2\cos\theta}{2} \right)^2-1}\ \bigg)d\theta.$

Hi I am trying to calculate this integral I given by $$ I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)d\theta. $$ ...
1
vote
1answer
29 views

Integrating an equation with both cos and tan

$$\int2\cos^5x\cdot\tan^6x\cdot dx$$ $$2\int\cos^5x\cdot\frac{\sin^6x}{\cos^6x}\cdot dx$$ $$2\int \frac{\sin^6x}{\cos{x}} dx$$ $$2\int\cos^{-2}x\cdot \sin^6x\cdot \cos{x}\cdot dx$$ ...
0
votes
0answers
21 views

Integration over rectangles

"Prove that a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is integrable over every rectangle in $\mathbb{R}^n $ if and only if it is integrable over every ball in $\mathbb{R}^n$" So I'm stumped ...
0
votes
1answer
60 views

Integral of $\sin|x|$

$$\int\sin|x|~dx$$ We have two cases: x less than zero, or x equals or higher than zero. $$\int_{-\infty}^0\sin(-x)~dx+\int_0^\infty\sin x~dx$$ Left side of this sum is equals to right side, so we ...
0
votes
2answers
43 views

Integrating $g: ℝ^2\to ℝ$ - Order of Integration

The problem: My work: I found the two integrals to be equal to each other, which is clearly not the desired result. Any suggestions/pointers? Thanks!
0
votes
3answers
28 views

How to get from $3\int_{-1}^0 (x^3-x) dx \,\,\,- \,\,\, 3\int_0^1 (x^3-x) dx$ to $6\int_{-1}^0(x^3-x)dx$?

Homework problem: Set up the definite integral that gives the area of the region. Two functions are given: $y_1 = 3(x^3-x)$ $y2 = 0$ The graph of $y1$ runs from x=-1 to x=1. I've gotten this ...
0
votes
2answers
44 views

How do you find the derivative of the $\int_{1-x}^{1+2x} e^{t^2} dt$?

$$\int_{1-x}^{1+2x} e^{t^2} dt$$ I don't fully understand the steps taken to answer this question. If someone could please tell me the steps for this kind of a situation I would really appreciate it. ...
2
votes
1answer
85 views

Simple integral $\displaystyle\int \frac{e^x}{x^2-a^2}\ dx$

Is this integral solvable? $$\int \frac{e^x}{x^2-a^2}dx,\quad a>0.$$
0
votes
2answers
46 views

Methods to do Integral

I know this integral can be done using complex analysis. Are there some slick solutions using standard calc methods? $\displaystyle\int_{-\infty}^{\infty}\displaystyle\frac{1}{(x^2+1)(x^2+9)}dx$
0
votes
0answers
6 views

Integration of characteristic function with varying boundaries

I'm a bit puzzled about integrals with indicator/characteristic functions in them. How do I start computing the following integrals? $$ A\int_{-\infty}^{\infty}f(x)\chi_{[-a+x,a+x]}dx $$ and $$ ...
2
votes
0answers
17 views

Question on the Prime Number Theorem (the Tchebychev Function)

This has been giving me nothing but a headache: Let the Tchebychev Function, $\psi (x)$ be defined: $$\psi (x) = \sum_{p^m \le x}\log p \space \space \space , \space \space \space p \in \mathbb P$$ ...
2
votes
0answers
39 views

Is there a generalization of integration by parts?

here is what i concerned: there are $u(x)$ and $v(x)$ in the original integration by part formula, what if the integral involve with one more function $w(x)$. Second of all, i know that there are ...
1
vote
1answer
27 views

Bessel's integral, how to actually evaluate?

I am just about to study Bessel functions and I have recently seen one of its integral representations given by: $$ J_ \alpha (x) = \frac{1}{\pi} \int_0 ^ \pi \cos(\alpha \tau - x\sin\tau) d\tau - ...
0
votes
1answer
19 views

Changing the domain of integral

I am studying how we use polar substitution to solve double integrals. However, I am struggling with finding the correct limits of the transformed integrals to obtain a suitable solution. eg: Why ...
1
vote
1answer
66 views

Derive The Midpoint Rule: $\int_{x0}^{x1}f(x)dx=hf(x_0+\frac{h}{2})+\frac{h^3}{24}f^{2}(\mu)$

The Given Question is: ================================================================== Expand the function $f(x)$ in a $1^{st}$ degree Taylor series about $x_0 + \frac{h}{2}$ with ...
1
vote
4answers
62 views

How can I prove the integral?

Prove that $$ \int\frac{dx}{x(\log_e x)^{7/8}} = 8(\log_e x)^{1/8} $$ I am totally lost on this subject. Any help how to prove this is appreciated!
6
votes
3answers
652 views

How do I solve this definite integral?

$$\int_0^{2\pi} \frac{dx}{\sin^{4}x + \cos^{4}x}$$ I have already solved the indefinite integral by transforming $\sin^{4}x + \cos^{4}x$ as follows: $\sin^{4}x + \cos^{4}x = (\sin^{2}x + ...
1
vote
2answers
33 views

Integrating a Partial Derivative

Would I be right to think that $$\int dx \,\,\,\frac{\partial}{\partial x} f(x,y)=f(x,y)$$ Or are there pathological cases?