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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1
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3answers
31 views

Integration by substitution exam question help

$$\int_1^2 x(2x-3)^4 \, dx\\ U = 2x - 3$$ I have rearranged to get $dx = dU/2$. So I am now at $\int xU^4 \, dU$ I am not quite sure what to do with the $x$ as it is not cancelled out as I thought ...
-4
votes
1answer
20 views

Solving Integral that includes radical expression 2

I need to solve this integral analytically. I used many methods but I can’t solve it. Please help me. Thank you
0
votes
0answers
10 views

Hydrostatic Force integration problem

Hello I was trying some work problems in my calculus textbook and came across one like this http://imgur.com/xdOKTht,cv41pvt#0 http://imgur.com/xdOKTht,cv41pvt#1 And I was wondering why the area was ...
-2
votes
2answers
31 views

Solving Integral that includes radical expression

I need to solve this integral analytically. I used many methods but I can’t solve it. Please help me. Thank you $$\int\sqrt{x^4-c}\ dx$$ http://i62.tinypic.com/15heux1.png
1
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0answers
16 views

Calculate the right Riemann sum to approximate the area of the region bounded by $f(x) = 25 - x^2$ on the interval $[-5, 5]$.

I'm attempting to calculate the right Riemann sum and approximate the area of the region bounded by $f(x) = 25 - x^2$ on the interval $[-5, 5] = [a, b]$. $$\sum_{k = 1}^{n}{f(a + k\Delta x)}\Delta ...
0
votes
1answer
18 views

Integrating a second order non homogeneous ODE

I took an exam and the teacher didn't solve this problem during the correction. I need to solve $$y''(x)-y(x)=\sin (e^x)$$I was able to find the solution to the homogeneous equation ...
4
votes
3answers
41 views

Evaluating $\int_{\sqrt{2}}^{\sqrt{5}} \frac{x^3}{\sqrt{x^2-1}} dx$ by substitution

$$\int_{\sqrt{2}}^{\sqrt{5}} \frac{x^3}{\sqrt{x^2-1}} dx$$ $u^2 = x^2 - 1$ I have worked out that $dx = du$ and that $u = x - 1$ so, $\int\frac{x^3}{u} du$ - but I'm stuck at this stage. Any ...
4
votes
0answers
34 views

What is the integral of $e^{a \cdot x+b \cdot y}$ evaluated over the Koch Curve

What is $$\int_{K} e^{a \cdot x+ b \cdot y} \mu(x,y)$$ where $K$ is the Koch curve and $\mu(x,y)$ is a uniform measure. Attempt: I can evaluate the integral numerically and I have derived a method ...
-5
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0answers
24 views

I need help on a few questions i have no idea how to answer [on hold]

If an object moves such that its velocity is given by m/s Use integration methods to find an equation giving the distance of the object at any time. AND The growth of a saw-tooth waveform flowing ...
-3
votes
2answers
38 views

Intergration the following functions with respect to $x$ [on hold]

$x^2\cos{x}$ $\ln{(x-1)}$ $(\ln{x})^2$ I know I must use the integration by parts to solve these questions but I have no idea at all how to continue.
0
votes
1answer
12 views

Prove that the image of a curve has zero content

Definition: A set $A \subset \mathbb{R}^2$ is said to have zero content if, for all given $\varepsilon >0$, exists a finite collection of rectangles $A_1, \dots, A_n$ such that $A \subset ...
0
votes
1answer
37 views

integration of $\frac{1}{\ln(A\times x^{0.5} + B)}$

I try to find solution for that: $\frac{1}{\ln(A\times x^{0.5} + B)}$ where $A$ and $B$ are constants. I don't mind to make some assumptions or expend it, but it doesn't work with Taylor expansion. ...
0
votes
0answers
19 views

How CES function with integral becomes min function in the limit

I wonder how a CES function over a continuum of goods, $$\left(\int_1^\infty c(\theta)^\delta g(\theta) \mathrm{d}\theta\right)^\frac{1}{\delta}$$ where $c(\theta), g(\theta)>0 \forall ...
0
votes
1answer
13 views

How has this answer been derived? Integration problem

I just don't understand how my teacher has got from part 1). to 2). Where has the 1/5 come from? Q(t) = ∫(1/((t^2)−t−6))dt ...
-2
votes
1answer
26 views

Integration using Substitution [on hold]

Firstly, I know that the graph of function, $f$ must cut the x-axis at least once such that the definite integral will equal to zero so I can apply Roelle's theorem somewhere. For b (i), letting $u ...
1
vote
1answer
14 views

How do boundaries change in this particular double integral?

Is there a simpler representation for an integral of the form $$\int_1^x \int_1^t f(u)\; du\; dt$$ analogous to $$\sum_{t=1}^x \sum_{u=1}^t f(u)=\sum_{t=1}^x (x-t+1)f(t)$$ ? It seems like there should ...
1
vote
1answer
30 views

Integration Properties

I have always had a mental block towards this property and would be truly grateful if someone would please help me. $$\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$$ Consider $$f(x) = x, for ...
1
vote
1answer
62 views

Integration with Limits

Find $$\displaystyle \lim_{n \to \infty} \int^{1}_{0}(x^{n}+(1-x)^{n})^{\frac{1}{n}}dx$$ Now, the answer is $$\dfrac{3}{4}$$ Now, the solution was hinted like this: using the property ...
1
vote
0answers
17 views

An example in which the Fubini theorem is inapplicable

This is example 8.9(a) in Rudin's Real and Complex Analysis, (alternatively, exercise 10.2 in Rudin's Principles of Mathematical Analysis). Let $X$ and $Y$ be the closed unit interval $[0,1]$, let ...
1
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2answers
71 views

Proving that $ f: [a,b] \to \Bbb{R} $ is Riemann-integrable using an $ \epsilon $-$ \delta $ definition.

Problem. Show that a bounded function $ f: [a,b] \to \Bbb{R} $ is Riemann-integrable if and only if for every $ \epsilon > 0 $, there exists a $ \delta > 0 $ such that for any partition $ ...
0
votes
0answers
27 views

Area of a surface of revolution about the y-axis-

I'm trying to find the area of a surface of revolution generated by the curves $$y=x^3,\quad x=1,\quad x=2, \quad\rm{around} \quad y=-1 $$ \begin{array}{lcl} A &=& 2\pi \int_1^2 {(y + 1)\sqrt ...
2
votes
1answer
28 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 ...
3
votes
2answers
49 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
22 views

Gauss Chebyshev formula [on hold]

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
24 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
31 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
21 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 ...
4
votes
1answer
55 views

Solve integral with exponent

How to solve integral: $$\int^\infty_0e^{-\frac{At^2}{t+1}}~dt , \quad A>0$$
-3
votes
1answer
36 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
43 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
83 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
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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
1answer
24 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
77 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
81 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
68 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
14 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
55 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
49 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
139 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
40 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
57 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
29 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 ...