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
87 views

Improper integral $\int_{0}^{\pi} \frac{x}{\sin x} dx$

Find out whether or not the following integral exists $$\int_{0}^{\pi} \frac{x}{\sin x} dx.$$ I'm pretty sure this integral doesn't exist but I can't seem to find a good way to prove this. It ...
1
vote
3answers
90 views

Evaluate $\int \frac{1}{(2x+1)\sqrt {x^2+7}}dx$

How to do this indefinite integral (anti-derivative)? $$I=\displaystyle\int \dfrac{1}{(2x+1)\sqrt {x^2+7}}dx$$ I tried doing some substitutions ($x^2+7=t^2$, $2x+1=t$, etc.) but it didn't work out.
0
votes
0answers
35 views

The negative integral meaning

Whenever I take a definite integral in aim to calculate the area bound between two functions, what is the meaning of a negative result? Does it simly mean that the said area is under the the x - axis, ...
1
vote
0answers
63 views

Integration Error

Sorry if this doesn't make any sense or if I did something obviously wrong, I was just playing around with taylor series' and then I got stuck. I know from the geometric series that: ...
1
vote
2answers
40 views

Euler's method for first three approximations?

I have tried variations of the problem for an hour at least and cannot get around to sloving this one. Thank you for input!
3
votes
1answer
33 views

Volume of a solid(between two planes)?

A solid lies between planes perpendicular to the y-axis at $ y=0$ and $y=1$. The cross-sections perpendicular to the y-axis are circular disks with diameters running from the y-axis to the parabola ...
1
vote
1answer
38 views

Integration: substitution then differentiation result different to differentiation then substitution.

I want to simplify this derivative ($n$ is an integer) $$ \frac{d}{d\theta} \int_{0}^{2\pi} e^{i n \phi} e^{i 2\pi k r \cos(\phi - \theta)} d\phi $$ If I substitute $\psi = \phi - \theta$ and ...
1
vote
1answer
32 views

Length of a curve?

I know how to find arc length and set up the equation in normal circumstances, but I have failed in all attempts to even set up this problem. I cannot even find a good example similar to this to get ...
2
votes
2answers
39 views

Shell method to find the volume of a solid?

Region bounded by $y=3x-2$, $y=\sqrt{x}$, and $x=0$ about the $y$-axis. I have been doing the washer method for all of my problems up to this one, and cannot seem to find a good resource to help guide ...
0
votes
1answer
36 views

Line integral calculation

Problem Let $f:[-1,1] \to \mathbb R$ be a $C^1$ function such that $f(-1)=0=f(1)$ and $f>0$ on $(-1,1)$. Knowing that the graph of $f$ is contained in the set $\{(x,y) : x^2+y^2\leq 1, y\geq 0 ...
5
votes
1answer
124 views

Hard integral, low hints… [duplicate]

$$\int_{ - \pi /2}^{\pi /2} \frac1{2007^{x} + 1}\cdot \frac {\sin^{2008}x}{\sin^{2008}x + \cos^{2008}x} \, dx .$$ This integral stuns me for a while, I just can't solve it! I tried integration by ...
5
votes
2answers
99 views

Is $\int^x \cos \frac1t$ differentiable at zero?

From Spivak's Calculus, 4th ed., exc 14-20: Let $$f(x) = \begin{cases} \cos \frac1x, & x\neq 0\\ 0, &x=0. \end{cases}$$ Is the function $\int_0^xf$ differentiable at zero? I'm having ...
0
votes
2answers
63 views

How to prove $ \int \sqrt{a^2-u^2}du $

How can I prove that the following definite integral? $$ \int \sqrt{a^2-u^2}du = \frac{1}{2}\left[u\sqrt{a^2 - u^2} + a^2 \arcsin\left(\frac{u}{a}\right)\right] +C$$
0
votes
1answer
22 views

Not getting limits of integration right (triple integrals)?

Calculate the volume bounded by $y+z=1$, $z=x^2-1$, $z=1-x^2$ and $y=0$ So our volume $V=\int \int \int _D1 dA$, all that's left is figuring out limits of integration. One method I saw was looking at ...
-4
votes
0answers
26 views

Triple intergrals with Volume [closed]

This question is a continuation of number 2 from the questions requiring deeper understanding of this homework. Suppose that you do not know the values of $a, b$, and $c$. Generalize what you did in ...
0
votes
1answer
70 views

Show that $\int_{\pi/4}^{\pi/2} \frac{\sin x}{x}\,dx\leq \frac{\sqrt{2}}{2}$

Show that $$\int_{\pi/4}^{\pi/2} \dfrac{\sin x}{x}\,dx\leq \dfrac{\sqrt{2}}{2}$$ Any Ideas, how to start ?!
-1
votes
1answer
37 views

Proving the indefinite integral $ \int \frac{1}{u^2(a+bu)}du $ [closed]

How can I prove that the indefinite integral $$ \int \frac{1}{u^2(a+bu)}du $$ is equal to $$ -\frac{1}{a}\left(\frac{1}{u}+\frac{b}{a}\ln\left|\frac{u}{a+bu}\right|\right)+C\ ? $$
2
votes
4answers
135 views

What is the integral of x/ln(x)?

Well, I'm french so excuse me if I make some mistakes in english... I have to calculate this integral : $$ \int_{e}^{2e} \frac{x}{\ln(x)} dx $$ But I don't know how, can you help me please? Thank ...
0
votes
1answer
27 views

Euler's method and Riemann sum

For: $F(0) = 0$ and $F'(x) = f(x)$ Euler's method: $F(0+h) = F(0)+ hF'(0) = 0 + hf(0)$ Continuing the process, $F(10h) = hf(0)+hf(h)+hf(2h)+.....hf(9h)$ This resembles the Riemann sum: ...
5
votes
1answer
48 views

Maximum value problem

A function $\hspace{0.1cm}$$f:[0,1]\to[-1,1]$$\hspace{0.1cm}$ satisfying$\hspace{0.1cm}$ $|f(x)|\leq x$$\hspace{0.1cm}$ $\forall x\in[0,1]$. Then find the maximum value of: ...
0
votes
1answer
26 views

Newton-Cotes Quadrature formula

Im trying to find more information about numerical integration methods. When is a Newton-Cotes Quadrature formula on n nodes exact?
2
votes
1answer
36 views

Upper bound the integral or its PV (or prove that it diverges)

I need help in finding an upper bound the following integral (or its Cauchy Principal Value): $$ \int_0^1 \sqrt{x} \frac{|\ln(\frac{(y^{-1}-1)}{(x^{-1}-1)})|}{|x-y|} dx $$ This integral arises as ...
0
votes
1answer
20 views

Finding the boundaries of integration when calculating P(X + Y > a) or P(X + Y < b) (Jointly Distributed Continuous Random Variables)

I have a problem on setting the boundaries of integration when I'm trying to find probabilities like $P(X + Y > a)$ or $P(X + Y < b)$. For example, when I have $f(x,y) = \frac {x} {5}\ +\frac ...
6
votes
3answers
242 views

Integral $\int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$

Calculate the following integral: \begin{equation} \int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx \end{equation} I am having trouble to calculate the integral. I ...
4
votes
4answers
110 views

If $f(x)$ is discontinuous at $x=0$, can $\int_{-1}^1 f(x)dx$ exist.

I am interested in the reasoning. All help is appreciated
-1
votes
2answers
67 views

Prove that $\displaystyle\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$

Prove that $\displaystyle\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$ Note that $c$ need not belong to $(a, b)$ And $f(x)$ is a continuous function. All ideas are appreciated.
1
vote
1answer
21 views

Double integral calculation where $x=(y-1)^{2}-1$ and $y=x$. Not sure whether I should do it in terms of $y$ or $x$?

This is what it looks like: My first strategy was to separate it into two by drawing a vertical line at x=0 and calculate the first half in terms of x first, and the second half in terms of y ...
7
votes
2answers
207 views

Another integral for $\pi$

Here is a new integral for $\pi$. $$\int_{0}^{1}\sqrt{\frac{\left\{1/x\right\}}{1-\left\{1/x\right\}}}\, \frac{\mathrm{d}x}{1-x} = \pi $$ where $\left\{x\right\}$ denotes the fractional part of ...
1
vote
1answer
73 views
5
votes
3answers
156 views

Examples of “difficult” integrals with are easier to solve with a series?

Yesterday someone posted an extremely elegant solution to a seemingly bizarre series where the integral: $$\int_{0}^{1} x^{m}\ dx = \frac{1}{m + 1}$$ was utilized. Oftentimes one will also ...
7
votes
1answer
136 views

Proving that a function is analytic

I'm struggling with the following problem: Problem: Suppose that $h$ is a continuous function on a simple closed curve $\gamma$. Define $$ H(w) = \oint_{\gamma} \frac{h(z)}{z - w} \, dz. $$ Show ...
5
votes
3answers
108 views

How to $\int_{0}^\infty {\sin^3(x)\over x}dx$

How to evaluate : $$\int_{0}^\infty {\sin^3(x)\over x}dx$$ I don't know how to do it. I tried to finish it using integration by parts, but it doesn't work? Can someone tell me how to evaluate the ...
2
votes
3answers
48 views

Parametrizing curve for complex analysis integral

I'm trying to show that $$ \int_{|z-z_0| = R} (z-z_0)^m \, dz = \begin{cases}0, & m \neq -1 \\ 2\pi i, & m =- 1. \end{cases} $$ Here's my attempt at a solution: We parametrize the curve at ...
2
votes
1answer
46 views

Proving the indefinite integral $ \int\frac{\sqrt{a+bu}}{u}du $

How can I prove that the indefinite integral $ \int\frac{\sqrt{a+bu}}{u}du $ is equal to $ 2\sqrt{a+bu} +a \int\frac{1}{u\sqrt{a+bu}}du$
2
votes
1answer
45 views

Defining the integral on an arbitrary metric space

I am trying to prove a version of Mercer's Theorem for an arbitrary compact metric space; that is, I do not wish to restrict myself to the space of real-valued continuous functions $C[a,b]$. I ...
2
votes
0answers
100 views

Definite trigonometric integral

This question is motivated by Iterative Mean, Covariance Algorithm Convergence: Is there a closed form for the integral $$ \int_0^{2 \pi} ...
3
votes
2answers
48 views

Comparison of Newton-Cotes Quadrature and Gaussian Quadrature formulas

Newton-Cotes quadrature formulas are a generalization of trapezoidal and Simpson's rule. The trapezoidal rule involves $2$ points, Simpson's rule involves $3$, and in general Newton-Cotes formulas ...
0
votes
5answers
53 views

Proving the indefinite integral $\int \frac{1}{u(a+bu)}du $

How can I prove that the definite integral: $\int \frac{1}{u(a+bu)}du $ is equal to: $\frac{1}{a} ln|\frac{u}{a+bu}| +C $
1
vote
0answers
36 views

Question related to Stokes theorem proof

I was reading Stokes theorem proof from Apostol's Calculus II textbook and I got stuck at some parts of the proof. I'll copy the parts where I had doubts: Suppose $S$ is a surface parametrized by ...
4
votes
5answers
185 views

How is $\,\int(1/x)\,dx = \ln|x|\,$ true?

Why does the integral of $\frac 1x dx$ equal the natural log of the absolute value of $x$? $$\int \frac 1x\,dx = \ln|x| + C$$
1
vote
2answers
40 views

Jordan measure zero discontinuities a necessary condition for integrability

The following theorem is well known: Theorem: A function $f: [a,b] \to \mathbb R$ is Riemann integrable if and only if its set of discontinuities has Lebesgue measure zero. Now if we change ...
1
vote
1answer
72 views

Proof about Riemann integrability of a bounded function

I tried to prove the following, please could somebody tell me if my proof is correct? If $f: [a,b]\to \mathbb R$ is a bounded Riemann integrable function then for every $\varepsilon > 0$ there ...
0
votes
1answer
40 views

Definite integral and Riemann sum

$ \int_0^1 x^2 dx $ n is the number of pieces to cut into For $ (x_i - x_{i-1}) = \frac {0+1}{n} $ and $ x_i = \frac in $ Using the right Riemann sum: $ \Sigma_{i=1}^n f(x_i)*(x_i - x_{i-1}) $ $ ...
0
votes
1answer
67 views

Integration Question $\frac{x^2 - 1}{x^4 + 3x^2 + 1}$

Is anyone able to check the answer to my integration question. $$ \int \frac{x^2-1}{x^4 + 3 x^2 + 1} dx. $$ EDIT: by using $$u = 1 + \frac{1}{x^2}$$ as a substitution Is the answer to the question ...
5
votes
1answer
188 views

How to find this integral $\int_{0}^{1}\ln\ln\bigl(1/x+\sqrt{(1/x^2)-1}\,\bigr)dx$ [duplicate]

How do I compute this integral ? $$I=\int_{0}^{1}\ln{\left(\ln{\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)}\right)}dx$$ In the math chatroom someone suggests setting ...
1
vote
3answers
48 views

integrating $\int_{\gamma}e^zdz$ with $\gamma$ is the arc on the unit circle that unites one with i

I am stuck integrating $$\int_{\gamma}e^zdz$$ with $\gamma$ is the arc on the unit circle that unites one with i. I tried this : The integrand $\mathrm{e}^z$ is holomorphic for $\vert z \vert \le ...
-1
votes
1answer
42 views

Value of line integral [closed]

Let $C=\{(x,y)\in \Bbb R^2:~\max\{|x|,|y|\}=1\}.$ The value of the line integral $$\oint_C(xy^2+2y+sin(e^x))dx+(x^2y+cos(e^y))dy$$ is?
0
votes
0answers
36 views

Definite integral with doable improper case

Is there a way to evaluate one or both of the following integrals: $$ \int_{a_1}^{a_2} e^{ib_1(x+b_2 \sqrt{1+x^2})}dx \quad \text{and}\quad \int_{a_1}^{a_2}\frac{x}{\sqrt{1+x^2}} e^{ib_1(x+b_2 ...
2
votes
1answer
37 views

Changing the order of integration (for Lebesgue-Stieltjes integral and Riemann integral)

Do the Lebesgue-Stieltjes integral and the Riemann integral have the same rules about the change of order of integration? I mean I know how to deal with Riemann integral, but I'm not sure if I can ...
0
votes
1answer
26 views

How to compute the area of this set in the plane?

Let $f$ be a non-negative function which is defined, bounded, and integrable on a closed interval $[a,b]$, and let $$ S \colon= \{\ (x,y) \ | \ a \leq x \leq b, \ 0 \leq y < f(x) \ \}. $$ Then is ...