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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1answer
13 views

Can a Function have Multiple Valid Indefinite Integrals

Working with U-substitution, I have to integrate the following. $\int x\cos(x^2)\sin(x^2)dx$ From my understanding you can take the integral by substituting $u$ for either $\cos(x^2)$ or ...
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0answers
8 views

$\frac{(m+1)^2} {\pi^2} \int_{-\infty}^{+\infty} \frac{1+(-1)^m \cos(\pi y)}{\left[y^2 -(m+1)^2\right]^2} dy = 1$ for $m\in\mathbb N$

I have an exercise that asks to prove: $$ \frac{(m+1)^2} {\pi^2} \int_{-\infty}^{+\infty} \frac{1+(-1)^m \cos(\pi y)}{\left[y^2 -(m+1)^2\right]^2} dy = 1 \quad \text{for}~ m\in\mathbb N$$ I also have ...
0
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1answer
26 views

Differentiating both sides with respect to time.

So I have this problem: An active volcanic mountain grows in the shape of a cone while maintaining its base diameter equal to its height. The volume of the mountain increases at a rate of ...
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1answer
17 views

How to compute $\sum\limits_{a=1}^{\infty}\int_0^b\lambda\left(\int_0^{\lambda}e^{-t}t^{a-1}dt\right)d\lambda$

Please suggest an efficient method to compute the following integral \begin{equation} I = \sum\limits_{a=1}^{\infty}\int_0^b\lambda\left(\int_0^{\lambda}e^{-t}t^{a-1}dt\right)d\lambda \end{equation} ...
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1answer
26 views

Computing a specific line integral

Here is the problem as I have been given it: A curve $C$ is given in Cartesian coordinates by $r(t) = (cos(sin(nt))cost,\; cos(sin(nt))sint,\; sin(sin(nt)))$, with $t$ between $0$ and $2$$\pi$ ...
2
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3answers
36 views

Integral $\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \ln(1+c\sin x) dx$, where $0<c<1$

I am trying to evaluate the following integral: $$\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \ln(1+c\sin x) dx,$$ where $0<c<1$. I can't really think of a way to find it so please give me a hint.
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2answers
27 views

Level curves for “unsolvable” integral

Problem: Sketch the level curves of g defined by $$g(x,y)=\int_x^y{e^{-t^2}dt}$$ (The error function does not need to be used here). Attempts at solution: (1) Apparently we could take $y=x$, then ...
2
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1answer
29 views

Integral reducing

I'm not following all the steps to this integral. I understand the exponent manipulation between steps 1 and 2, but I don't understand how the integral goes from step 2 to step 3. (i.e. I don't ...
0
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1answer
46 views

How to find the integral $\sin^2\sqrt2x$

I need help finding the integral of $\sin^2\sqrt2x$ I started to integrate it using integration by parts: $u=sin^2\sqrt2x$ and $dv=dx$ $\int u \,{\rm d}v = uv - \int v\,{\rm d}u$ But ...
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1answer
27 views

Area of Lemniscate of Bermoulli

I need to find out area of one loop of Lemniscate $r^2 = \sin(2\theta)$. I have tried taking square root and substitution but those haven't led to anything.
3
votes
3answers
83 views

Evaluate the sum $x + \frac{x^3}{3} + \frac{x^5}{5} + … $

Evaluate the sum $$x + \frac{x^3}{3} + \frac{x^5}{5} + ... $$ I was able to notice that: $$ \sum_{n=0}^\infty \frac{x^{2n-1}}{2n-1} = \sum_{n=0}^\infty \int x^{2n-2}dx = \lim_{N\to\infty} ...
1
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1answer
30 views

Surface integral of $A:=\{(x,y,z)\in \mathbb{R}^3|x^2+y^2+z^2\leq 4, x\leq0,z\leq0\}$ using parametrization

Calculate the surface integral of $A:=\{(x,y,z)\in \mathbb{R}^3|x^2+y^2+z^2\leq 4, x\leq0,z\leq0\}$ using a suitable parametrization and the corresponding surface element. I think this set is a ...
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0answers
4 views

Conditional Probability in Multivariate Normal

Given a tri-variate Normal, the conditional probability of an element given others truncated information is Now if I know that the mean vector u is (-0.91,-1.31,-1.39) and R is ...
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1answer
20 views

calculate riemann integral on the triangle [on hold]

Calculate Riemann Integral $$ \int \int_{B} e^{4y^2}dxdy $$ where B is a triangle: (0,0), (0,1), (-1,1) I have no idea what should I do with that integral.
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4answers
33 views

Prove the inequalities without calculating the integrals

$$ \int_{0}^{\frac{\pi}{2}} \sin^4x dx \le \int_{0}^{\frac{\pi}{2}} \sin^3xdx$$ I have tried to define 2 functions $ f, g:[0, \frac{\pi}{2}] \rightarrow \mathbb{R}$ and say that $ f(x) = \sin^4x$ ...
-1
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0answers
7 views

Integrate $(\frac{y}{R})^{3/7}\, dA$

How do I find the integral for: $\displaystyle \bigg(\frac{y}{R}\bigg)^{3/7}\, dA$; where $R =$ pipe radius, $r = $radius from centerline, and $y = R-r$ ? I know I'm supposed to integrate from $y=R$ ...
0
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0answers
17 views

Calculate the integral of unknown geometric shape [on hold]

This is a 10m long tunnel for walkers. The profile is parabolic. How would you calculate the whole concrete needed for this tunnel? My problem is the geometric shape. :(
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2answers
63 views

$ \int \frac{dx}{4x^2-12x+13}$

This is probably not too hard but i can't get it right: I am trying to calculate $$\displaystyle \int \frac{dx}{4x^2-12x+13}$$. The solution is $\displaystyle ...
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3answers
53 views

Use integration by substitution

I'm trying to evaluate integrals using substitution. I had $$\int (x+1)(3x+1)^9 dx$$ My solution: Let $u=3x+1$ then $du/dx=3$ $$u=3x+1 \implies 3x=u-1 \implies x=\frac{1}{3}(u-1) \implies ...
-4
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0answers
26 views

For the following integrals find a and find b [on hold]

In the following picture, what is a=? what is b=?
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2answers
44 views

Integrating: $ \;\int \frac{1}{x^2+3x+2} dx $

How can I solve the following integral: $$ \int \frac{1}{x^2+3x+2} dx $$ Should I proceed by changing the variable (substitution)? or should I use integration by parts? Or another method ...
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2answers
22 views

Convergence of $\int_0^1 \frac{dx}{(\cos(x)-1)\sqrt{1-x^2}}$

Study the convergence of $$\int_0^1 \frac{dx}{(\cos(x)-1)\sqrt{1-x^2}}$$ Well, we can observe the $$\left| \frac{1}{(\cos(x)-1)\sqrt{1-x^2}} \right| \le \left| \frac{1}{(\cos(x)-1)} \right|$$ ...
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0answers
34 views

Problem about limit of an integral

I came across this question while doing some exercises on integrals, and I was wondering if I could get some help. a) Show that for $n < -1$, $\int_1^N x^n dx$ converges as $N \to\infty$, and for ...
0
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1answer
42 views

Estimating the integral $\int \frac{\sin(x)}{x}\, dx$. [on hold]

Would anyone be able to help me out with this question? I'm not quite sure how to go about it. Thanks in advance! Consider the integral $$ I = \int_{\pi/2}^\pi \frac{\sin x}{x}\,dx. $$ This integral ...
1
vote
1answer
30 views

Find the derivative of an integral.

Find the derivative of the following integral $$ F(x)=\int_x^{x^2}e^{t^7}dt $$ Find F′(x) given F(x). Normally I would show my attempt in working out the problem: however, I don't even know where ...
2
votes
1answer
88 views

Proof in Hamilton: Divergence theorem for differential forms?

For a vector field $X\in\Gamma(TM)$ on a closed Riemannian manifold $(M,g)$ we have \begin{align*} \int_M\text{div}X\;\mu=0, \end{align*} where \begin{align*} ...
1
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1answer
53 views

How to integrate $1/\sqrt{(1+x^2)^3}$?

Normally I use WolframAlpha pro to help me with problems I don't know however wolfram wont/cant show me the steps only the final solution to this integration problem. Is anyone able to assist me with ...
1
vote
1answer
34 views

Simple Trig Integration. Why is my answer wrong?

$$\int \dfrac{\cos x+\sin 2x}{\sin x}dx=\int \dfrac{\cos x+2\cos x\sin x}{\sin x}dx=\int \dfrac{\cos x\left(1+2\sin x\right)}{\sin x}dx$$ Substitute $u=\sin x$ and $du=\cos x\ dx$: ...
3
votes
1answer
47 views

Riemann sum $\int x^m dx$?

I'm trying to find the Riemann integral of $x^m$ between $a$ and $b$ with $b>a$. So far I have managed to get $$\int_a^b x^m~dx=\lim_{n\rightarrow \infty}\left(a^m \times \frac{b-a}{n} \times ...
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0answers
21 views

Show that $\int_a^b f(x) dx=\lim_{n\rightarrow \infty} \sum_{k=0}^{n-1} \int_{x_k}^{x_{k+1}} f(x) dx$.

I've come up with a proof for the following statement, but I'm not quite sure it's 100% correct. I would appreciate any help: If $f$ is integrable on $[a,b]$, $x_0=a$, and $x_n$ is a sequence of ...
0
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0answers
19 views

How to evaluate $\int_{-\infty}^t v(\alpha)\,d\alpha$ for $v(\alpha) = \sin\alpha$?

Integral form of the inductor's V-I relation is: $$i(t) = \frac{1}{L} \cdot \int_{-\infty}^t v(\alpha)\,d\alpha$$ How can I determine this function for $v(\alpha) = \sin\alpha$? I've tried to but I ...
0
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0answers
25 views

The base of S is the triangular region with vertices (0, 0), (3, 0), and (0, 2). Cross-sections perpendicular to the y-axis are semicircles.

Find the volume of the following solid S: The base of S is the triangular region with vertices (0, 0), (3, 0), and (0, 2). Cross-sections perpendicular to the y-axis are semicircles. So far I got ...
1
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0answers
23 views

exponential integration with fractional powers

I am trying to solve the following integral $$\int_{-\infty}^a \frac{\beta_1 \beta_2}{y^2(c-y)^2} e^{-\beta_1/(c-y)} e^{-\beta_2/y} \, dy$$ where $a<0$, $c>0$, $\beta_1>0$, $\beta_2>0$ I ...
1
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2answers
39 views

Study the convergence of $\int_1^\infty \frac{x\ln x}{x^4-1} dx$

Study the convergence of $\int_1^\infty \frac{x\ln x}{x^4-1} dx$ So first we have two potentially problematic points which are $1,\infty$ We split the integral to $$\int_1^2 \frac{x\ln x}{x^4-1} ...
1
vote
3answers
49 views

convergence of $\int_a^b \frac{1}{x^2} dx$

Why is it true that $\int_0^a \frac{1}{x^2} = \infty$ but $\int_a^\infty \frac{1}{x^2} \lt \infty$? Shouldn't it be symmetric?
1
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1answer
28 views

Solving integral (by substitution?)

How do I solve the integral $\int \frac{1}{\sqrt{b-x^2}}$ where b is a constant ? I know that $\int \frac{1}{\sqrt{1-x^2}} = \arcsin(x)$ , so I guess I have to substitute somehow clever. Can you ...
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0answers
17 views

Simple proof for a continuous-time linear system and impulse $\delta$?

From Schaum's Outlines of Signals & Systems: Let's work with continuous-time signals. Let $T$ be a linear time-invariant system (LTI). Input $x(t)$ can be expressed as $x(t) = ...
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1answer
33 views

How one can compute $\int_0^1 \frac{dt}{t} t^{\pm \frac{1}{2}} \exp[ -x(t+t^{-1})] = \sqrt{\frac{\pi}{x}} e^{-2x}$? [on hold]

For positive $x$ \begin{align} \int_0^1 \frac{dt}{t} t^{\pm \frac{1}{2}} \exp[ -x(t+t^{-1})] = \sqrt{\frac{\pi}{x}} e^{-2x} \end{align} How to compute this integral?
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1answer
37 views

Divergence and convergence of the integral. [on hold]

I have the following integral, $$I=\int_a^b |x|^{-p} dx$$ where $a<b$ are finite real numbers and $p\leq 0$ is a non-negative real number. If we start solving it we will come up with the ...
2
votes
2answers
40 views

Evaluate the integral $\int_0^{\ln(2)} \sqrt{(e^x-1)}dx$

Evaluate the integral $\int_0^{\ln(2)} \sqrt{(e^x-1)}dx$ Why is it wrong to... $$\int_0^{\ln(2)} \sqrt{(e^x-1)} dx= \int_0^{\ln(2)} (e^x-1)^{1/2} dx= \frac{2}{3}(e^x-1)^{3/2} |_0^{\ln(2)}$$
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0answers
24 views

Proof that there is no identity to integral operation on any set of functions

The statement is: Let $f\in F, f:x\mapsto f(x)$ be a function($F$ contains sufficiently non-trivial functions). Then $\not\exists I\in F$, so that $$\int_{-\infty}^\infty If=f(0)$$ What I am implying: ...
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0answers
33 views

Riemann sum/integral of $\sin(x)$ from $0$ to $A$ [duplicate]

Hello I keep getting stuck on calculating the Riemann sum/integral of $\sin x$ from $0$ to $A$ I know this has been looked at before but I just don't understand it and was hoping someone could ...
0
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2answers
72 views

how to evaluate the integral $ \int_0^{2\pi} \frac{\sin{nx}\cos{nx}}{\sin{x}}dx$?

would someone give me a hint or a solution ? how to evaluate the integral $ \int_0^{2\pi} \frac{\sin{nx}\cos{nx}}{\sin{x}}dx$? Thanks a lot.
4
votes
1answer
45 views

Calculation of Radon–Nikodym derivative

Suppose the function $X \colon \mathbb{R} \longrightarrow \mathbb{R} \colon x \longmapsto X(x) : = x^2$. I want to calculate the Radon–Nikodym derivative $\frac{\text{d}\lambda_X}{\text{d}\lambda}$, ...
0
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0answers
56 views

Integrate $x e^{-x^4}$ Advanced Calculus

I'm having trouble integrating the equation below. The book says "Hint: Let $u=x^2$, etc." I don't know how I would let $u=x^2$ since in order to do integration by parts I need $u\,dv$ and we only ...
0
votes
2answers
105 views

Find the antiderivative of $(x^2+x+1)^{20}$ [on hold]

How do I find the antiderivative for that? The online calculators say that there's no solution. **NVM, the little scratch on my paper turned out to be a faintly copied handwritten $'2'$, turning it ...
-2
votes
0answers
20 views

Definite integrals and piecewise defined functions [on hold]

Consider the function $G(x) = \int_0^x g(u)\, du$ , where: $ g(u) = \begin{cases} 4 - \frac 43 u, & \text{for $0 \leq u < 6$} \\ u - 10, & \text{for $6 \leq u \leq 12$}. \end{cases} $ i. ...
1
vote
2answers
116 views

For what values is my integral diverging or converging?

Is the following integral convergent $$\int_{\gamma}^{+\infty} \left(1-\dfrac{1}{1+sv^{-1}}\right)\left(\frac{1}{\alpha_1}v^{\frac{2}{\alpha_1}-1} \, e^{-\beta\, v^{\frac{1}{\alpha_1}} }+ ...
0
votes
1answer
74 views

My integral is behaving strangely

The following integral is something I am trying to solve for $$ \int_{\gamma}^{\infty} \Bigg[ 1- \left( \frac{2a(1+s x^{-1})+b}{1+s x^{-1} } \right) \Bigg] x^{\frac{2}{\alpha}-1} \, dx$$ We have ...
3
votes
1answer
46 views

Volume of a sphere with two cylindrical holes.

Consider a sphere of radius $a$ with 2 cylindrical holes of radius $b<a$ drilled such that both pass through the center of the sphere and are orthogonal to one another. What is the volume of the ...