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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Inequality $\int^{1}_{0}(u(x))^2\,\mathrm{d}x \leq \frac{1}{6}\int^{1}_{0} (u'(x))^2\,\mathrm{d}x+\left(\int_{0}^{1} u(x)\,\mathrm{d}x \right)^2$

I've been scratching my brain on this one for about a week now, and still don't really have a clue how to approach it. Show that for $u \in C[0, 1]$ the following inequality is valid: ...
0
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1answer
17 views

I don't understand one of the steps in solving Green's function for diffusion

Why is it that $\int_0 ^\infty u^4e^{-u^2}du = $ $ \left[\frac {d^2}{d\alpha^2}\int_0 ^\infty e^{-\alpha u^2}du \right]_{\alpha = 1} $?
4
votes
2answers
39 views

$\int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$

Good evening! I want to compute the integral $\displaystyle \int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$. However I find it extremely difficult. What I've tried is rewritting it as: ...
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0answers
30 views

Find $\int \tan(\tan x)\hspace{1mm}dx$

Find $\int \tan(\tan x)\hspace{1mm}dx$ This is an Interesting problem, which I have been trying from different directions, nothing seems to work, its been a day on this one. Can anyone figure out ...
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3answers
69 views

How would you integrate $\sqrt{1+\frac{1}{x^2}}$

I need to integrate $\sqrt{1+\frac{1}{x^2}}$ I've tried to let $u=1/x^2$ but end up with, $\int \frac{\sqrt{1+u^2}}{u^{3/2}}du$ . I attempted to then substitute $u=\tan\theta$ and lead me to ...
0
votes
3answers
38 views

Integration by Parts [Answer Provided].

The function: $$f(x) = x^2/e^x$$ Perhaps, someone could show how the solution becomes: $$\int f(x)dx=-e^{-x}(x(x+2)+2)$$ It is likely that my integration by parts contains an error.
2
votes
4answers
43 views

Second order homogenous non-linear DE: $3(x')^2 - 2x''x=0$

How do I solve this for $x$? $$3\dot{x}^2-2\ddot{x}x=0$$ $$\Leftrightarrow$$ $$3(x')^2 - 2x''x=0 $$ Note: This comes from my working here(on stack exchange meta sandbox[newest activity]) List of ...
3
votes
2answers
41 views

Closed form for $\int z^n\ln{(z)}\ln{(1-z)}\,\mathrm{d}z$?

Problem. Find an anti-derivative for the following indefinite integral, where $n$ is a non-negative integer: $$\int z^n\ln{\left(z\right)}\ln{\left(1-z\right)}\,\mathrm{d}z=~???$$ My attempt: ...
4
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0answers
34 views

Closed form for ${\large\int}_0^1\frac{\ln^{\color{magenta}3}x}{\sqrt{x^2-x+1}}dx$

This is a follow-up to my earlier question Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$. Is there a closed form for this integral? ...
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0answers
14 views

Moment Generating Function of a nonlinear transformation of an exponential random variable.

Let $\tau$ be an exponential random variable, with parameter $\lambda$. Let $$ V = \delta^\tau $$ where $0 < \delta <1$. Sorry if this notation seems strange, but it is what I am using, I ...
-1
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1answer
27 views

How to derive laplace formula for given integral of laplace transform? [on hold]

As stated in title how to derive formula directly from definition? $$ \int_0^ \infty te^te^{-st} dt $$
2
votes
3answers
49 views

How to simplify this double integral using substitution?

I want to take this integral: $$\int_{0}^{L}\int_{0}^{L} |x-y| \,dx\,dy$$ And set $u = |x-y|$ and convert this integral into something that only needs one integral. I don't know the correct way to ...
3
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0answers
33 views

How to solve integrals of the form $\int u^{-\alpha} e^{-\beta u} du$?

I have to simplify an integral of the form $\int u^{-\alpha} e^{-\beta u} du$, where $\alpha, \beta \in \mathbb{R}^{++}$. Is it a standard integral, or a family that subsumes gamma integrals? Is there ...
2
votes
3answers
80 views

Find $\int \dfrac{dt}{t-\sqrt{1-t^2}}$

Find $\int \dfrac{dt}{t-\sqrt{1-t^2}}$ MY APPROACH : Substitute $t = \sin x$ Multiply numerator and denominator by $\cos x+\sin x$ then rewrite everything in terms in $\sin2x$ and $\cos2x$, we ...
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1answer
30 views

how to prove the only difference between antidrivaties of a function is in their constants?

how to prove "If F is an antiderivative of f on an interval I , then the most general ...
2
votes
1answer
51 views

How to find limit of this integral???

I had this problem on my last exam,and I couldn't do it: $$\begin{align} \lim_{x \rightarrow \infty}\frac{\int_{0}^{x} e^{t^{2}}dt}{x^{5}\int_{0}^{x^{2}}\frac{e^{t}}{t^{2}} dt} \end{align}$$ ...
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0answers
49 views

Does this integral have a closed form?

$I=\int_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx$ I saw this question here. It is really hard to find a closed form. Or is there no closed form? Please give me a hand. Thanks!
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votes
0answers
55 views

Is there a way to rewrite integrals in Mathematica using u substitution? [migrated]

In Mathematica, can I give it an integral and a few substitution rules and have it rewrite the integral in terms of those variables?
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2answers
37 views

Evaluate a Nested Integral

According to two questions [1] and [2] asked on this site earlier there exists a nice relation: $$\frac1{n!} \left(\int_{0}^t\mathrm dt \; f(t)\right)^n = \int_{0}^t\mathrm dt_1 \int_{0}^{t_1}\mathrm ...
2
votes
1answer
43 views

How to simplify this complex integral? [on hold]

How to approximate this integral as a function of a and b? $$\int_0^\pi\int_0^{2\pi}\sqrt{(a-b\sin\varphi\cos\theta)^2+(b\cos\varphi)^2+(b\sin\varphi\sin\theta)^2}d\theta d\varphi$$ where a and b ...
4
votes
2answers
59 views

Green's function for y'' + y = f(x)

This example is taken from the Wikipedia's article. Namely, find the Green's function for $$y'' + y = f(x)$$ with boundary conditions: $$y(0) = y(\frac {\pi} {2}) = 0.$$ The defining equation for ...
1
vote
1answer
28 views

Volume of a solid bounded by surfaces - is it correct?

Could you check if my calculations and reasoning are correct. And maybe suggest a nicer way of solving this problem? We are given a solid bounded by these surfaces: $y=x^2, \ y=1, \ 2x+y+z = 4, \ ...
0
votes
1answer
49 views

Differential and Integral calculus.

Can anyone here explain me, why do we take the Centre of mass of a conical shell using slant height and $dl$ whereas the centre of mass of a solid cone is calculated using the vertical height and ...
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0answers
45 views

Integral of $\displaystyle \int \sin^{-1} \left (e^{\sqrt x}\right ) \mathrm{d}x$ …

How do I evaluate the following; $$\displaystyle \int \sin^{-1} \left (e^{\sqrt x}\right ) \mathrm{d}x$$ $$\displaystyle \int \sin^{-1} \left (e^{-\sqrt x} \right )\mathrm{d}x$$ Is there a closed ...
3
votes
4answers
216 views

Finding the definite integral of a function that contains an absolute value

The integral in question is this: $\int_{-2\pi}^{2\pi}xe^{-|x|}$ My attempt: Since there is a modulus, we split it up into cases. I'm not really sure which cases to split it into, do I just ...
0
votes
1answer
22 views

Derivative of integral over part of Gaussian distribution

I am currently trying to compute the following derivative and integral: $$ P\psi_\theta = \frac{d}{d\theta}\int_{-k}^k tf_T(t)dt, $$ where $t=x-\theta$ and $X\sim N(\theta_0,\sigma^2)$. $f_T$ above ...
0
votes
1answer
18 views

An integral of Wolstenholme:$\int_0^{+\infty}\frac{\sum_1^n A_k\cos{a_k x}}{x}\mathrm {d} x$ where $\sum A_k=0$ and $a_k>0$

The book by Whittaker and Watson says it's equal to $-\sum_{k=1}^n A_k \log {a_k}$, and attibutes it to Wolstenholme. I believe this readily reduces to the simpler case of evaluating $\displaystyle ...
0
votes
3answers
50 views

Solve integral $\int_0^\pi \sin(t-2nt)dt$

I have the integral $$\int_0^\pi \sin(t-2nt)dt$$ Wolfram states that the answer is: $$ \frac{2\cos^2(n\pi)}{1-2n} $$ But I can't get the same... I am close to the answer with the calculation below, ...
0
votes
2answers
27 views

Proof of the substitution rule for integrals for the indefinite case

I know that the substitution rule works like this: By the chain rule: $$F(g(x))' = f(g(x))g'(x)$$ Then $$\begin{align} \int_a^b F(g(x))'dx &= \int_a^b f(g(x))g'(x)dt\\ F(g(b)) - F(g(a)) ...
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2answers
37 views

Find $f$ such that $f'(x)=ax^2+bx$, given the values of $f'(1)$, $f''(1)$, and $\int_0^2 f(x)\,dx$

The question is : Find the solution $f(x)$ if $f'(x)=ax^2+bx$, and (i) $f'(1)=6$, (ii) $f''(1)=18$, (iii) $\int_0^2 f(x)dx=18$. My solution is: According to $(i)$, we know $6=a+b$, and ...
0
votes
1answer
26 views

How to prove that this equality is the development of a fourier series?

how can I show that this identity is a development of a fourier series? $$f(x)=\sin^3 x=\frac{3}4 \sin x-\frac{1}4 \sin 3x$$ I tried this: obtain the Fourier coefficients whih $$b_n=\frac{2}\pi ...
3
votes
2answers
73 views

Weird integral symbol : $\mathrel{\int\!\!\!\!\!-}$

What does this integral sign mean ($\int$ with line going through the middle)? $$ \mathrel{\int\!\!\!\!\!\!-} $$ (It had something to do with the Beckenbach-Radó Theorem)
1
vote
1answer
22 views

Volume of revolved solid using shell method: finding height

The problem that I am working with is: Find the volume of the solid of revolution formed by rotating the region $R$ bounded by $y = 4+ x^2,\;x=0,\;y=0,\;and\;x=1$ about the line $y=10$ I have ...
0
votes
1answer
35 views

Integrals involving roots

I am bit stucked with an integration form while doing one of my proofs for a graphics application.Issue is I cant take out the terms from the trigonometric functions for a proper known integral ...
0
votes
3answers
31 views

Improper Integrals

Determine whether the following improper integral is convergent or divergent. $$\int_1^{\infty} \text{sech}\, x \ln x \,dx$$ I think that I need to use integration by parts but the sechx is really ...
2
votes
3answers
348 views

Is the reverse of the second fundamental theorem of calculus always true?

The second part of the fundamental theorem of calculus states that if $$ f(x)=g'(x) $$ then $$ \int^b_a f(x)\,dx = g(b)-g(a)$$ And so I was wondering that if you solved $\int^b_a f(x)\,dx$ using some ...
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0answers
19 views

Finding the volume of a solid from revolution

Revolve $y=4+x^2$ bounded by $x=0,$ $x=1,$ and $y=0$ around $x=8$ I have started by splitting the area in two regions and using the shell method to get the part between y=4 and y=5 with: ...
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2answers
23 views

convolution problem given $h(x)=1/2$ for $0<x<2$ and $0$ otherwise

I have a convolution problem in the form $$g(x)= \int_{-\infty}^\infty h(y)h(x-y)\,dy$$ where they give me the function $h(x)=1/2$ for $0<x<2$ and $0$ otherwise. I have never done a ...
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votes
1answer
28 views

Existence of Monotone Sequence of Simple Functions

Let $\Omega$ be a measurable space with measurable sets $\Sigma$ and denote the space of simple functions by:$$\mathcal{S}:=\{s:\Omega\to\mathbb{R}:s=\sum_{k=1}^K ...
0
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1answer
41 views

When does this integral converge?

So I'd like to find out for which values of $a,b>0$ the following integral is well-defined and how that will change if the absolute value is removed? Thanks! $I = \int_0^\infty ...
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2answers
24 views

I need to find the formula for h(x) for all x

we're given a function $h(x)=\begin{cases}1/2&\text{for}&0\le x<2\\0 &\text{otherwise}\end{cases}$. Then we are told to define the function $\displaystyle g(x)= ...
0
votes
2answers
40 views

How to evaluate a definite integral that contains a constant?

How to evaluate this definite integral with a constant? $$\int_0^{a^1/4} x^7\sqrt{a^2 - x^8} dx$$ I've never seen the constant 'a' before in this situation. But here's what a have so far at least ...
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votes
2answers
31 views

Find $\int t\sin^{-1}t\hspace{1mm}dt$

Find $\int t\sin^{-1}t\hspace{1mm}dt$ How do we approach this question, is there a simple way to integrate
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votes
3answers
73 views

Find $\int (\arcsin x)^2\hspace{1mm}dx$ [on hold]

Find $\int (\arcsin x)^2\hspace{1mm}dx$ $ $ How do we approach this problem
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1answer
24 views

Measurability of Modulus

Context: This problem came up while reading an essay on Bochner integrability. Let $\Omega$ be a measure space and $E$ a Banach space. Consider two plain functions $f:\Omega\to E$ and $g:\Omega\to ...
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2answers
29 views

How to find the derivative of improper integral with variable upper limit?

I have the integral from $-\infty$ to $y^2$ of the function $(e^{-|x|})$ and I need to find the derivative of this. That is, $$\frac{d}{dy} \int_{-\infty}^{y^2} e^{-|x|}\,dx$$ Usually derivative ...
2
votes
2answers
49 views

Integrating $ 3 \displaystyle \int \frac{\cos^5 x}{\sin x} dx$?

I need to solve the integral $$3 \int \frac{\cos^5 x}{\sin x} dx$$ I'm not sure how to proceed from here. I think Integration by Parts may be useful, but I'm not entirely sure what to make $u$ and ...
1
vote
4answers
55 views

Integration by parts of $\frac{\ln(1+\sqrt{x})}{\sqrt{x}}$

(i) Use the substitution $u=\sqrt{x}$ to show that $$\int_0^1 \frac{1}{1+\sqrt{x}} dx = 2-\ln4$$ (ii) Use integration by parts to show that $$\int_0^1 \frac{\ln(1+\sqrt{x})}{\sqrt{x}} dx = ln16 ...
1
vote
1answer
15 views

double integral initial condition

For an initial condition problem, if I take the first integral: $\int1 dt = t + C1$ But if I take the 2nd integral, do I end up with: $\int (t + C1)dt = \frac{1}{2} t^2 + C1t + C2$ or does the ...
1
vote
1answer
47 views

Rotate about the x-axis with respect to dy

How would I rotate the region bounded by $y = 4+x^2,\;x=0,\;y=0,\;and\;x=1\;$ along the x-axis in terms of dy. I have already solved this problem in terms of dx see here Here is the ...