Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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2
votes
1answer
16 views

equality between variable and integral

I received the following question as part of my homework: Let $f(x)$ be a continuous function onto $[0,1]$. $f(x)\le\frac{1} {2\sqrt{x}}$ for every $0<x\le1$. Prove that x=0 is the only solution ...
1
vote
2answers
30 views

Find the following indefinite integral: $\int (x^2+6x+5)^{12} (x+3) \ dx$

The solution I got was $(1/13)(x^2+6x+5)^{13} + C$ I am not sure if I am correct though and would like help. Thanks!
-2
votes
0answers
14 views

Gauss Chebyshev formula

Use Gauss Chebyshev formula with $n=3$ to approximate the value of the integral. $$\int \frac{x^4}{\sqrt{1-x^2}}dx$$ from -1 to 1. Also compare the result with true value, where the zeros and the ...
1
vote
2answers
23 views

Changing order of double integral

I have a double integral with the integral with respect to x on the inside between 0 and y^2 and the outer integral with respect to y between 0 and 1. If i change the order of the integrals what would ...
0
votes
1answer
24 views

Evaluate $\int_\gamma z^ne^{1/z}dz$, where $\gamma$ is the unit circle.

I need to evaluate $\int_\gamma z^ne^{1/z}dz$, where $\gamma$ is the unit circle traveled in the counterclockwise direction. I'm thinking about writing the function as a Laurent series and then ...
0
votes
1answer
20 views

Integration of step functions

I've managed parts (a) and (b) fairly easily, but c is causing me a real headache. I've seen the Cauchy-Schwartz inequality before, but I've hit a roadblock because I've no idea whether or not I can ...
2
votes
0answers
28 views

Solve integral with exponent

How to solve integral: $$\int^{+\infty}_{0} e^{-At^2/(t+1)} \, dt , \quad A>0$$
-3
votes
1answer
31 views

What is $\int_0^2\int_{y/2}^{(y+4)/2}y^2(2x-y)e^{(2x-y)^2}dxdy$? [on hold]

What is $$\int_0^2\int_{y/2}^{(y+4)/2}y^2(2x-y)e^{(2x-y)^2}dxdy$$ if we change the region of integration to a rectangle?
2
votes
1answer
40 views

Calculate area of the region formed by $f(x)= x^3-x^2$ and x-axis

What is the area of the region formed by the graph of $f(x)=x^3-x^2$ and the $x$-axis in the interval $[0,3]$? Did I do this right? I get $$\int_0^3x^3-x^2\,dx$$ giving me the answer of $45/4 = ...
2
votes
0answers
77 views

Finding $\int \frac{\sin\sqrt{\frac{x}{2}}}{\sqrt{x\cos\sqrt{x}}}dx$

Finding $$\int \frac{\sin\sqrt{\frac{x}{2}}}{\sqrt{x\cos\sqrt{x}}}dx$$ This is a homework. I tried to solve it by assuming $x=u^2$ but after that the integrals become not simple, so I don't know how ...
5
votes
2answers
56 views

An integration question to be solved without using differentiation under the integral sign.

$$I(\alpha)=\int_0^1 \frac{x^\alpha-1}{\ln x}dx.$$ As the title says, if someone could solve this without using the differentiation under the integral sign technique, I would be very grateful.
1
vote
1answer
26 views

write $\iiint_E \hspace{1mm}dV$ in 6 forms. where $E = \left\{ (x, y, z)|0\leq z\leq x+y, x^2\leq y\leq \sqrt{x},0\leq x\leq 1\right\}$

write $\iiint_E \hspace{1mm}dV$ in 6 forms. where $E = \left\{ (x, y, z)\hspace{1mm}|0\leq z\leq x+y, x^2\leq y\leq \sqrt{x},0\leq x\leq 1\right\}$ As you can see two forms are easy. $$\iiint_E ...
0
votes
0answers
14 views

The maximum value (peak) of multiple self-convolution of rectangular function

In Multiple self-convolution of rectangular function - integral evaluation, formula for self-rectangular function of rectangular function seems to have been derived. How do we prove that this formula ...
4
votes
3answers
72 views

How do I compute this integral?

I'm wondering how to compute the integral $$ \int_2^3\int_0^\sqrt{3x-x^2}\frac{1}{(x^2+y^2)^{1/2}}\,\mathrm{d}y\mathrm{d}x. $$ Clearly it is too complicated to do it directly, so I'm guessing you have ...
4
votes
2answers
76 views

Prove the Dirac Delta Function satisfies $ x\frac{\mathrm{d} \delta(x)}{\mathrm{d} x} = -\delta(x) $

$ x\frac{\mathrm{d} \delta(x)}{\mathrm{d} x} = -\delta(x)$ I've been told that this answer involves integration by parts. I began like this: $\int x\frac{\mathrm{d} \delta(x)}{\mathrm{d} x} = ...
3
votes
2answers
57 views

Proving that the delta function is the derivative of the step function.

I want to prove $\frac{\mathrm{d} }{\mathrm{d} x}\Theta =\delta (x)$ using this representation of the delta function: $\delta(x)= \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{ikx}dk $ This should be ...
0
votes
0answers
13 views

Fining the angular bounds of a triple integral function

This problem requires the taking of a triple integral over a region. I believe it's most useful to convert to cylindrical coordinates, which I did. However, I could not find the theta bounds due to ...
1
vote
3answers
52 views

Evaluate the definite integral $ \int_{\pi/6}^{\pi/2} \frac{\cos(x)}{\sin^{5/7}(x)}\, dx$

Evaluate the integral: $\displaystyle \int_{\pi/6}^{\pi/2} \frac{\cos(x)}{\sin^{5/7}(x)}\, dx$ (using substitution) Here's my attempt at solution: u = $\sin^5(x)$ $du = 5\sin^4(x) \cdot \cos(x) ...
-2
votes
1answer
27 views

Computing definite integral with u-substitution [on hold]

How to compute $$\int_{0}^{\sqrt{3}} \frac{dx}{\sqrt{4-x^2}}$$ and $$\int_{1}^{2} \frac{dx}{3+x^2}$$ using only $u$-substitution?
0
votes
2answers
32 views

Evaluate the indefinite integral $\int \frac{1}{x^2} \sin\left(\frac{6}{x}\right) \cos\left(\frac{6}{x}\right) \, dx $

Evaluate the indefinite integral: $\displaystyle \int \frac{1}{x^2} \sin\left(\frac{6}{x}\right) \cos\left(\frac{6}{x}\right) \, dx $ (using substitution) The answer is: $\frac {1}{24} ...
4
votes
0answers
47 views

Trigonometric integral of $f(x)=(x^2)(\sin(x^2))$. [duplicate]

I've tried with the chain rule and $u$-subtitution ($u=\sqrt{x}$) but I get nothing. Can you help me please? $$\int (sqrt{x})(\sin(x)) \ dx$$
2
votes
3answers
41 views

Evaluate $\int \frac{\sec(11 x) \tan(11 x)}{\sqrt{\sec(11 x)}} \, dx $

Evaluate the indefinite integral: $\displaystyle \int \frac{\sec(11 x) \tan(11 x)}{\sqrt{\sec(11 x)}} \, dx $ (using substitution) The answer is: $\frac{2}{11} \sqrt{sec(11 x)} + C$ I don't get ...
0
votes
0answers
22 views

Minimizing Unintegrable Exponential Function

I am trying to develop an algorithm which minimizes an unintegrable function. I don't have a strong mathematics background and am unaware of such strategies. My integral is of the following form: ...
2
votes
5answers
134 views

Estimate $\int^1_0 e^{-x^2}\, dx$

Estimate $\int^1_0 e^{-x^2}\, dx$ This is in a section on Taylor series so I would assume that is how it should be solved. I started by using the Taylor series formula for $e^x$ replacing $x$ with ...
0
votes
0answers
18 views

Difference between measure zero and volume zero?

I have the following definitions for a set to have measure zero and for a set to have volume, respectively: A set $A$ has measure zero if for any $\epsilon > 0$ there is a covering $\{S_i\}_{i \in ...
2
votes
4answers
39 views

Integral of $\frac{e^x}{5+2e^x}$

Regarding the integral of this term$\frac{e^x}{5+2e^x}=\frac{e^x}{2(\frac{5}{2}+e^x)}$ Is the answer $\frac{1}{2} \ln(\frac{5}{2} +e^x)$ or $\frac{1}{2} \ln(5+2e^x)$? When I substitute $u= ...
0
votes
1answer
25 views

Using Taylor series find derivatives of arctan(x)

Using Taylor series for $arctan(x)$, find $f^{(5)}(0)$ and $f^{(6)}(0)$ for $f(x)=arctan(x)$ I figure for this problem I compare the general Taylor series formula to the Taylor series for $arctan(x)$ ...
0
votes
2answers
55 views

How do i evaluate the following integral?

Hi I was wondering if someone can help me evaluate the following integral. Show that if $-1 < x < 1$, then $$\int_{0}^{\pi} \frac{\log{(1+x\cos{y})}}{\cos{y}}dy= \pi \arcsin{x} $$ thank you ...
3
votes
1answer
28 views

Find $\lim_{n \to \infty} n^{\alpha} \int_{n}^{\infty} \frac{f(x/n^2)}{x^{\alpha + 1}}(x-n)dx$

I am looking at an old exam in my measure theory and integration class. I am trying to solve a problem and am wondering if I am doing it right. Problem Let $f$ be a bounded measurable function on ...
2
votes
0answers
44 views

On a problem about Rolle's theorem

Let $f:[1,3]\to\mathbb R$ be a continuous function such that $\int_1^2 f(x)dx=2$, and $\int_1^3 f(x)dx=3$, then there exists a real number $c\in(2,3)$ such that $$ \int_1^c f(x)dx=cf(c) $$ Note. I ...
0
votes
1answer
42 views

How do we prove $\int \frac{\ln(1+x)}{x}dx = -\sum_{k=1}^{\infty}\frac{(-x)^k}{k^2}$?

After working on the integral $\int_{0}^{1} \frac{\ln(1+x)}{x}dx$ for a couple of hours, I became convinced its antiderivative was not elementary. So I looked it up on Wolfram Alpha, and it found that ...
3
votes
2answers
58 views

Hint on how to find $\int \frac{x^2}{1+x^2}dx$

I am almost sure that this would have been asked before, but how can one find $$ \int \frac{x^2}{1+x^2} dx? $$ If I had a $x^2 - 1$ in the denominator, then I could factor into $(x-1)(x+1)$ and use ...
2
votes
0answers
22 views

Double Integral of an Exponential Function with an Absolute Value in the Numerator of the Exponent

This is a question related to statistics, but my major concern relates to the setup and evaluation of integrals. So I decided this question was better suited for Mathematics Exchange than CV. I know ...
3
votes
3answers
103 views

Evaluate $\lim _{n\to \infty }\int_1^2\:\frac{x^n}{x^n+1}dx$

We have $$I_n=\int _1^2\:\frac{x^n}{x^n+1}dx$$ and we need to find $\lim _{n\to \infty }I_n$. Have any ideea how we can evaluate this limit?
0
votes
3answers
25 views

Given that: $T(x,y)=\ \int_{x-y}^{x+y} \frac{\sin(t)}{t}dt\ $, calculate: $\frac{\partial T}{\partial x}(\frac{\pi}{2}, - \frac{\pi}{2})$.

Given that: $T(x,y)=\ \int_{x-y}^{x+y} \frac{\sin(t)}{t}dt\ $, How do I calculate: $\frac{\partial T}{\partial x}(\frac{\pi}{2}, - \frac{\pi}{2})$? I seriously have no direction for how to solve ...
1
vote
2answers
49 views

Find the integral: $\int x^{7/2} sec^2(2+x^{9/2}) \mathrm{d}x$

Find the integral: $\int x^{7/2} sec^2(2+x^{9/2}) \mathrm{d}x$ Can I multiply and distribute the $ \ x^{7/2}\ $ and $ \ sec^2 \ $ together. What is the strategy to solve this problem.
1
vote
1answer
18 views

Initial value problem through origin

$\frac{dz}{dt}=8t*e^z$, Through the origin I have never done an initial value problem before, but I took it to mean that it gave me the initial value of the differential equation (0, 0) and that I ...
3
votes
0answers
19 views

Stieltjes Integral - If $f, f^2, g, g^2\in R(\alpha)$ for an arbitrary integrator $\alpha$, then is $fg\in R(\alpha)$

My question is if $f, f^2, g, g^2\in R(\alpha)$ on $[a,b]$ for an arbitrary integrator $\alpha$, then is $fg\in R(\alpha)$ as well? This question stemmed from a problem in Apostol's Analysis, in ...
1
vote
2answers
26 views

Find solution to the differential equation

$\frac{dB}{dx}+2B=50$ $B(1) = 50$ I tried separating the variables but that didn't work, and without separating the variable I'm not sure what to do.
0
votes
3answers
44 views

Integral of $\cosh^3(x)$

What is the integral of $\cosh^3(x)$? And how exactly can I calculate it? I've tried setting $\cosh^3(x)=(\frac{e^x+e^{-x}}{2})^3$ but all I get in the end is one long fraction.
1
vote
1answer
13 views

Showing something involving integrals is an inner product

I have this problem: Let $C([0,1])$ be the real vector space of continuous functions on the interval [0,1]. Show that $<. , .>: C([0,1]) \times C([0,1]) \rightarrow \mathbb{R}$ ...
2
votes
1answer
46 views

Integrate $\int_{0}^1 (1 + 4y^2)^{1/2} dy$ [duplicate]

$$\int_{0}^1 (1 + 4y^2)^{1/2} dy$$ So, how do I integrate this without the use of trigonometrical substitution? Can anybody give me a hint? Thank you!
2
votes
1answer
74 views

Trouble solving an integral

So I have been trying to solve this equation, The given answer is, I began by using substitution to change the integral. Substituting t back in where t is taken from 0 to infinity. ...
7
votes
1answer
130 views

Poisson Integral relation

If $$ I_n(r) = \int_0^\pi \frac{\cos nx}{r^2-2r\cos x+1} \, dx $$ How to prove that $$ I_{n-1}(r)+I_{n+1}(r)= \left(r+\frac{1}{r}\right)I_n(r)\text{ ?}$$ I only find that $$I_{n-1}(r)+I_{n+1}(r)= ...
0
votes
0answers
17 views

Quality of approximation of an Ito integral

How could I investigate whether $$P(t,T-t)\left[a(T-t-\Delta)-a(T-t)+ (b(T-t-\Delta)-b(T-t))'x(t)+ \frac{1}{2}b(T-t)'\sigma\sigma'b(T-t)+ b(T-t)'(x(t+\Delta)-x(t))\right]$$ is a good or bad ...
2
votes
1answer
39 views

A question on integration of differential forms on a manifold

I'm fairly new to differential geometry and have been reading up on integration on manifolds. All the texts/lecture notes that I've read so far always consider integrating an $n$-form over an ...
2
votes
2answers
37 views

Need help solving complicated integral $\oint_{\mathcal C}\begin{pmatrix}x_2^2 \cos x_1 \\ 2x_2(1+\sin x_1)\end{pmatrix} dx$

Let $\mathcal C$ be the curve that traces the unit circle once (counterclockwise) in $\mathbb R^2$. The starting- and endpoint is (1,0). I need to figure out a parameterization for $\mathcal C$ and ...
1
vote
1answer
38 views

Understanding a particular transformation of an integral given in a proof

Using the theorem of mean values find the sign of the integral... $$\int_{0}^{2 \pi}{\sin x \over x}dx= \int_{0}^{\pi}{\sin x \over x}dx+\int_{\pi}^{2 \pi}{\sin x \over x}dx$$ Then: $[x-\pi=t ; ...
1
vote
3answers
44 views

Evaluate the integral $\int \frac{x}{(x^2 + 4)^5} \mathrm{d}x$

Evaluate the integral $$\int \frac{x}{(x^2 + 4)^5} \mathrm{d}x.$$ If I transfer $(x^2 + 4)^5$ to the numerator, how do I integrate?
1
vote
2answers
58 views

Show that the antiderivative exist [on hold]

I am new to this. How do I show that the antiderivative exist and show that is continuous too? Thanks