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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4
votes
1answer
80 views

Solving integral $ \int \frac{x+\sqrt{1+x+x^2}}{1+x+\sqrt{1+x+x^2}}\:\mathrm{d}x $

there is integral $$ \int \frac{x+\sqrt{1+x+x^2}}{1+x+\sqrt{1+x+x^2}}\:\mathrm{d}x$$ i am trying to separate this : $$=\int \mathrm{d}x -\int \frac{\mathrm{d}x}{1+x+\sqrt{1+x+x^2}} $$ but have no idea ...
1
vote
0answers
17 views

difference of the values of a function is an integral

This is a very simple quesiton but something I don't understand. From Taylor expansion: $$f(y)-f(x)=f'(x)(y-x)+O((y-x)^2)$$ so, if I just picture that, on the left is the difference between two values ...
3
votes
2answers
47 views

Evaluate $\int\frac{\mathrm{d}x}{2x-4}$

My question is to evaluate: $$\int\frac{\mathrm{d}x}{2x-4}$$ Why is the solution equal to $\frac{1}{2}\ln|x-2|$ as opposed to $\frac{1}{2}\ln|2x-4|$? I understand that if I factor $\frac{1}{2}$ ...
3
votes
2answers
75 views

Evaluate $\int x \sqrt{1 - x^4} \,\mathrm{d}x$

I have the following question $$\int x \sqrt{1 - x^4} \,\mathrm{d}x$$ I know we have to use trig. substitution for this and therefore, I did the following by letting $x = \sin \theta$ and $dx = \cos ...
1
vote
0answers
13 views

Prove with Lebesgue’s Criterion for integrablility that the composition $f\circ g$ is integrable

I have this homework question regarding Lebesgue's criterion for integrability and could use a bit of help. I'm not sure if my proof is entirely correct or formal enough. Here is said question: ...
0
votes
1answer
26 views

Evaluate $\lim\limits_{x\to\infty}\frac{1}{\sqrt{x}}\int_1^x\ln(1+\frac{1}{\sqrt{t}})dt$

$\lim\limits_{x\to\infty}\frac{1}{\sqrt{x}}\displaystyle\int_1^x\ln(1+\frac{1}{\sqrt{t}})dt=?$ If the limit exists with l'Hopital i get ...
3
votes
2answers
49 views

Integral of $\frac{1}{x^2+1}$ using complex partial fractions.

Is there any way to evaluate the following integral via a complex partial fraction decomposition? $$ \int \dfrac{1}{x^2 + 1} \text{ d}x $$ So far I have: $$ \begin{aligned} \int \dfrac{1}{x^2 + 1} ...
6
votes
4answers
664 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?
1
vote
0answers
42 views

Cauchy Integral Theorem problem (lack of understanding)

First of all i was asked to evaluate this integral $\int_\gamma \frac{2z}{(z-1)(z-3)} dz$ where $\gamma (t) = 2e^{it}$ for $0\leq t \leq 2\pi$. Now I thought I would have to calculate this ...
0
votes
1answer
50 views

Floating Point Number System

I really have no idea of how to do these questions - in fact I have no idea of how to do any question in the paper - but I have tried to figure out what's going on in the course called Computational ...
7
votes
3answers
101 views

Showing that $\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$

I was reading an article in which it was stated that, with a change of variable, one could show that: $$\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$$ I tried with $t = 1 + ...
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 ...
2
votes
1answer
19 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
25 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 ...
-2
votes
1answer
40 views

Integrate $\cot^2x-\frac{\cos^2x}{\tan^2x}$ [on hold]

Integrate $\int{\cot^2x-\frac{\cos^2x}{\tan^2x}}dx$
-2
votes
1answer
45 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
31 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
72 views

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

My try, using $x = \sec(u)$ substitution: $$ \begin{eqnarray} \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 \\ &=& ...
-1
votes
0answers
23 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
48 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
96 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
42 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
42 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 ...
1
vote
0answers
30 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 $u=(x^2+2)^{1/2}, du=x(2+x^2)^{-1/2}dx$ ...
6
votes
0answers
93 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
vote
2answers
41 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
53 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
47 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
11 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 ...
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 ...
1
vote
1answer
21 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
22 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
26 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
55 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
225 views

William Lowell Putnam Integral Problem

Prove That $$ \frac{22}{7}-\pi= \int_0^1 \frac{x^4\,\left(1-x\right)^4}{1+x^2}$$
1
vote
2answers
131 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 ...
1
vote
1answer
30 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
61 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
39 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
18 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 ...
1
vote
1answer
46 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
43 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
66 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
30 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
45 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. ...