Questions tagged [integration]

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 the integrals being considered. This tag often goes along with the (calculus) tag.

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1answer
76 views

How to integrate $\sin{\theta}dt$ [closed]

I was doing some physics and I got on a problem which is how to integrate $\sin{\theta}dt$, but the problem is that $\theta$ is a function in terms of $t$ and we don't know what that function is. I ...
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1answer
54 views

Do I have to use integral for this problem? [closed]

Find the area of the shape formed by the $(x+y)^5=2021x^2y^2$ function graph.
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1answer
66 views

Integral notation in Shifrin's *Multivariable Mathematics*

On page 348 of Shifrin's Multivariable Mathematics, for a 1-form $\omega=\sum F_{i}dx_{i}$ on $\mathbb{R}^{n}$ and a parameterized curve $C$ given by a function $\mathbf{g}:[a,b]\rightarrow\mathbb{R}^{...
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2answers
62 views

Using Calculus to Solve for a Differential Equation

The question I have is: $x''(t)+yx'(t)+z^2x(t)=0$, where y and z $\in \mathbb{R}$ without $0$. $t \geq 0$ by the way. The total energy of the system is $$\frac{1}{2}(x'(t))^2+ \frac{1}{2}z^2x(t)^2$$ ...
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1answer
41 views

Limit of an definite integral

Calculate $$\lim\limits_{n\to\infty}\int_0^1 \frac{x^n(ax^2+ax+1)}{e^x}.$$ a) $0$ b) $a$ c) $2a+1$ d) $\dfrac{2a+1}{e}$ I tried to note separated $\displaystyle\int_0^1 \frac{x^n\cdot x^2}{e^x}$ with ...
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2answers
79 views

show that $f$ is Riemann integrable on $[0,1]$

Let $f(x)=\sin\left(\frac{1}{x}\right)$ if $0<x\le1$ and $f(x)=0$ if $x=0$. Show that $f$ is Riemann integrable on $[0,1]$ and calculate it's integral on $[0,1]$. I would like to know if my proof ...
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2answers
20 views

Solving differential equation with boundary conditions $y=2$ at $x=0$

I need to find a solution to this differential equation $(1+x^2) $$\tfrac{\mathrm{d}y}{\mathrm{d}x} = x-xy^2$, with boundary condition $y=2$ at $x=0$ I collect all terms on the RHS to get $$\tfrac{\...
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1answer
58 views

Need help with proof on trigonometric integrals

Question: Let $f:[0,\pi]\rightarrow\mathbb{R}$ be a continuous function. Show that, if $$\int\limits_0^{\pi} f(t)\sin(t)dt = \int\limits_0^{\pi} f(t)\cos(t)dt =0,$$ then the equation $f(x)=0$ admits ...
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40 views

Evaluating $ \int_0^{\Lambda}r^{d - 1}\log\left(1 + a\sqrt{r^2 + m_1^2} + b\sqrt{r^2 + m_2^2}\right)\,dr$

I'm working on quantum field theories, and when integrating out a field from the path integral I encounter the following integral: \begin{equation} I = \int_0^{\Lambda}r^{d - 1}\log\left(1 + a\sqrt{r^...
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Proving mean value theorem without Rolle's theorem

I started to wonder whether I could prove the mean value theorem without using Rolle's theorem by using limits and integration. My point was to see if I could prove the theorem by noting that it ...
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34 views

Airy functions - integral

I am struggling to simplify following integrals consisting of following Airy functions. $u=C_3 e^{-\beta_0 x_2} \int _0^{x_2} e^{2\beta_0 x_2} \int _0^{x_2} e^{-\beta_0 x_2} Ai(-\frac{(-i A Re \...
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0answers
47 views

How can we get the domain of $F(x) = \int f(x) dx$?

Let $$F(x) = \int f(x) dx.$$ My question is how can we get the domain of F(x) generally. I usually think in f(x), and I imagine where it makes sense to take the area under the curve, but I'm afraid ...
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24 views

Interval of which $S_1$ and $S_2$ are equal in volume.

Let the region $R$ lie above the curve, $y= 1 -ax^2$, and below $y = 1$, on $0\le x \le 1$, and $0 < a \le 1$. So $R$ lies entirely in the first quadrant. Let $S_1$ be the solid obtained by ...
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21 views

Question Involving Finding Volume Using Integrals

I have a calculus question concerning integrals here. For a region $R$ between a function (purely arbitrary, but with the restriction $f(x)>0$) $y = f(x)$ and the x-axis, on [$0$, $a$], a > 0. ...
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17 views

Quantitative criteria of Lebesgue Integrability on $\mathbb{R}^d$.

Suppose that $f : \mathbb{R}^d \to \mathbb{C}$ is a Lebesgue measurable function on $\mathbb{R}^d$. Then a famous results are following : if $f$ satisfies \begin{equation} |f(x)| \lesssim \frac{1}{|x|^...
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37 views

Find, with calculus shapes of function $y=f(x)$, $y = g(x)$

There's a question of integrals. The region between some unknown function, $y=f(x)$,$y = g(x)$, and $f(x) \geq g(x) \geq 0$ in intervals $[2,5]$ is rotated about the lines, $x=9$ to make a 3D shape. ...
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1answer
43 views

Laplace transform vanishing at all integers

I am looking to find a function $f$ which is not zero almost everywhere ($f\neq 0 ~a.e.$) and such that $$ \int_{\mathbb{R}} f(t) e^{-ts} dt = 0~\forall s\in\mathbb{Z}$$ I was thinking of taking an ...
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2answers
60 views

Evaluate this integral. [closed]

$$\int \frac{x^3}{\sqrt{36-9x^2}}dx$$ I've tried u-sub and trig sub but don't know which answer is correct.
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1answer
69 views

Fitting a derivative of a curve to data

Some derivatives cannot be integrated to an analytic function but have the function embedded in the derivative itself. I need to know how to fit those derivative functions to a set of data. Here is ...
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4answers
274 views

Prove $ \int_{5\pi/36}^{7\pi/36} \ln (\cot t )dt +\int_{\pi/36}^{3\pi/36} \ln (\cot t )dt = \frac49G $

I have the conjecture for the integral $$ \int_{\frac{5\pi}{36}}^{\frac{7\pi}{36}} \ln (\cot t )\>dt +\int_{\frac{\pi}{36}}^{\frac{3\pi}{36}} \ln (\cot t )\>dt = \frac49G $$ where $G$ is the ...
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1answer
46 views

For valid probability densities $p(x)$ and $q(x), \int p(x)\ln(q(x))~dx = 0$. Can you catch my mistake?

I was trying to solve the following integral by parts, $$ \int p(x)\ln(q(x))dx$$ for valid probability density functions $p(x)$ and $q(x)$, $$ \int p(x)\ln(q(x))dx = \ln(q(x)) - \int\frac{q'(x)}{q(x)}...
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1answer
30 views

Single variable Integration word problem

In 2000 the yearly world petroleum consumption was about 77 billion barrels and the yearly exponential rate of increase in use was 2%. How many years after 2000 are the world's total estimated oil ...
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1answer
21 views

Find two LI continuous differentiable functions $f,g: I \in \mathbb{R} \rightarrow \mathbb{R}$ such that $(f\cdot g')(t) - (f'\cdot g)(t)) = 0$.

Find two linear independent continuous differentiable functions $f,g: I \in \mathbb{R} \rightarrow \mathbb{R}$ with $(f\cdot g')(t) - (f'\cdot g)(t)) = 0$. Any idea on how can I find such functions? ...
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2answers
40 views

Integral beginner proof

Question: Let $f:[0,\pi]\rightarrow\mathbb{R}$ be a continuous function. Show that, if $$\int\limits_0^{\pi} f(t)\sin(t)dt =0,$$ then the equation $f(x)=0$ admits a solution in $[0,\pi].$ What I've ...
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1answer
32 views

Vector analysis on stokes theorem [closed]

How can I approach this kinda problem .Verify stokes theorem for $\vec{F} = z i + x j + y k$ . Curve is the one quadrant of the hemisphere $x^2 + y^2 + z^2 = 1$ .
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1answer
32 views

Why does this integral equals to given output?

I was reading about Gaussian distribution function: and I saw this formula: I don't understand why that integral is equal to $$\frac{\phi (t)}{t}$$ It seems like it must be equal to infinity. And ...
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2answers
51 views

How to prove (using integration by parts)

Could you help to prove this: $$Y(\lambda) = \int_{a}^{b} g(x) e^{i \lambda f(x)} \,dx$$ we should use integration by parts to get $$Y(\lambda) = \frac{1}{i \lambda }e^{i \lambda f(x)} \frac{g(x)}{...
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1answer
41 views

Calculating this series integral $\int (\frac{x^{2n-2}}{(2n-2)!})\cdot \frac{x^{2n-1}}{(2n-1)!}$

I've calculated this integral and would like the communities feedback and support towards my solution: $$\int \left(1 \cdot \frac{x^2}{2!} \cdot \frac{x^4}{4!} \cdots\frac{x^{2n-2}}{(2n-2)!}\right)\...
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1answer
34 views

How can I find the domain of $ \int_{0}^{x} \frac{\log(1-t^{2})+t^{2}}{t}dt$?

Let $$ h(x) = \int_{0}^{x} \frac{\log(1-t^{2})+t^{2}}{t}dt \ .$$ How can I find the domain of this function? I tried to integrate it, but it seems impossible; I also thought that it could be the ...
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1answer
45 views

What does the region enclosed by $y = 1/x^5 , y=0, x=3, x=4$ look like?

I'm having trouble trying to see what the region I'm supposed to be computing looks like. The volume of the solid obtained by rotating the region enclosed by $$y = 1/x^5 , y=0, x=3, x=4$$ about the ...
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0answers
67 views

Integration of $\int_a^b{f(x)^2 dx}$ for general $f(x)$?

I working on a math problem that requires an elegant, general solution for $$\int_a^b{f(x)^2\,\mathrm{d}x}.$$ I am struggling to find one and hope that somebody here can help out. By using integration ...
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2answers
48 views

Let $f(x)=1$ if $x=0$ and $f(x)=0$ if $x>0$. Show that $f$ is integrable.

Let $f(x)=1$ if $x=0$ and $f(x)=0$ if $x>0$. Show that $f$ is Riemann integrable on $[0,1]$. I think for everyone this question is really basic, but I'm just training myself on proving the ...
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0answers
49 views

How to evaluate given integral in terms of Gamma function?

I want to evaluate the below integral $$ \int_{0}^{\infty} e^{-ax^{2}-bx} x^{\alpha -1}\,\mathrm{d}x, ~~~~~~ \operatorname{Re}(\alpha)>0 $$ where $a,b$ are real constants. Somehow, I want to ...
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1answer
16 views

Integration of autocorrelation function when $\int_0^1 f(t) dt = 0$

When $\int_0^1 f(t) dt = 0$ and $f(t)=0$ for $t\in \mathbb{R} \setminus [0,1]$, my conjecture is that $$ \int_{\mathbb{R}} R_{ff}(\tau)\,d\tau = 0, $$ where $$ R_{ff}(\tau) = \int_{\mathbb{R}} f(t+\...
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3answers
87 views

Question regarding fixed point (integrals)

Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function. Show that, if $$\int\limits_0^1 f(t)\,\mathrm{d}t =\frac{1}{2},$$ then $f$ has a fixed point in $[0,1]$, i.e. $\exists x_0\in [0,1]$ such ...
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1answer
88 views

Show that $f$ is Riemann Integrable on $\left[0,\pi/2\right]$

Let $f(x)=\cos^2(x)$ if $x\in \mathbb{Q}$ and $f(x)=0$ if not. Show that $f$ is Riemann Integrable on $\left[0,\pi/2\right]$. The problem that I have, is that I don't really see why this function ...
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1answer
42 views

Determine the convergence or divergence of the sequence with $n$th term $n\pm(-1)^n$ [closed]

So I was given the following prompt when I was studying for a test: In the following sequence, determine the convergence or divergence with the given $n$th term. $\lim_{n \to \infty} n\pm(-1)^n$ I ...
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3answers
45 views

Proving that the $\chi^{2}$ Distribution is a PDF

The $\chi^{2}$-distribution is given by $$f(x; k)\equiv \frac{1}{2^{\frac{k}{2}} \Gamma\left(\frac{k}{2}\right)}x^{\frac{k}{2}- 1}e^{-\frac{x}{2}}.$$ If the $\chi^{2}$-distribution is a probability-...
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1answer
62 views

How can this integral be evaluated?

I am trying to find the definite integral of this function. When entered into wolfram alpha the result (shown below) is given. However I do not understand what the E (elliptic integral of the second ...
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0answers
34 views

Integrating Bessel's Function [closed]

Here a last year student, trying to program an analytical model for borehole heat exchangers. I have been researching for weeks on how to get a discrete solution of these equations, so that i can ...
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0answers
51 views

How should I calculate the following integral? [closed]

How should I calculate the following integral? $$\int_{-c}^{+c} \frac{\exp\left({-2\beta (a^2+y^2)^{1/2}}\right)}{\sqrt{a^2+y^2}}dy$$ $a$ and $\beta$ are constants and $[-c,+c]$ is the custom range.
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1answer
46 views

The convergence of the series after integrating term-by-term

I am interested in the integral $$ I = \int_{-\infty}^\infty e^{i A x} e^{- Bx^2 - \lambda x^4}, $$ for some real values $A$, $B>0$ and $\lambda >0$. One way to tackle this integral is to expand ...
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0answers
32 views

Evaluating integral $\int \frac{2x+7}{2x^2+x+3} \,dx$ [closed]

$$\int \frac{2x+7}{2x^2+x+3} \,dx$$ Any hints on how to begin?
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0answers
30 views

$\int_0^x \frac{t^n}{1-t} dt = ?$ [duplicate]

I want to calculate the integral, $\int_0^x \frac{t^n}{1-t} dt$, where $n \in \mathbb{N}$ and $x \in (0;1)$ are some constants, but I do not know how to do it. I tried the following methods: ...
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0answers
19 views

Is it possible to express an infinitesimal area as a sum of squares?

I came across a question which was asking the center of mass of an area, the centroid. The proposed solution uses weighted average, the formula is that given a planar region, the x and y coordinates ...
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0answers
22 views
2
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1answer
46 views

Compute $\int_{\gamma} z\, dz$ for any smooth path $\gamma$ which begins at $z_0$ and ends at $w_0$

Compute the complex line integral$$\int\limits_{\gamma} z\, \mathrm{d}z$$ for any smooth path $\gamma$ which begins at $z_0$ and ends at $w_0$. Confused as to how I am supposed to go about ...
2
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2answers
69 views

Find $\int_0^4(g\circ f\circ g)(x)\mathrm{d}x$ where $f(x)=\sqrt[3]{x+\sqrt{x^2+1/27}}+\sqrt[3]{x-\sqrt{x^2+1/27}}$, $g(x)=x^3+x+1$

Let $$f(x)=\sqrt[3]{x+\sqrt{x^2+\frac{1}{27}}}+\sqrt[3]{x-\sqrt{x^2+\frac{1}{27}}}$$ and $$g(x)=x^3+x+1$$ then, find $$\int_0^4(g\circ f\circ g)(x) \mathrm dx$$ My attempt: Let $\displaystyle h(x)=\...
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1answer
45 views

Integral of $\int\ln(a\sin(2x)+b)\cos(x)dx$

Mathematica is able to solve the indefinite integral $$\int\ln(a\sin(2x)+b)\cos(x)dx$$ analytically for $b>a>0$, but the resulting $\arctan$ terms are too complicated for the next steps I need ...
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2answers
76 views

What is the other way to write this triple integral?

I need to find all $5$ ways to write this integral and so far I have found $3$ of the $5$ but I need some help to find the last $2$ which are $dy$ $dz$ $dx$ and $dy$ $dx$ $dz$. I assume one of them ...