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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3
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2answers
107 views

How do I evaluate the following integral $\int_{-\infty}^\infty\mathop{dx} \frac{x^n}{(x^2+a^2)^m}$?

I am interested in the following integral $$\int_{-\infty}^\infty\mathop{dx} \frac{x^n}{(x^2+a^2)^m},$$ given that $m>n/2$ (this is just what I wrote so that the integral converges. If this is not ...
0
votes
1answer
43 views

Integration by parts (trig integral)

Evaluate the following integral: $$\int \cos(x)\cos(5x)\, dx$$ In the answers it says to solve by integration of parts twice with a consistent $u$ and $\frac{dv}{dx}$ to get ...
0
votes
2answers
26 views

Solve this integral using an appropiate substitution: $\iint_R x+y+1 dydx$

Solve this integral using an appropiate substitution: $$\iint_R x+y+1 dydx$$ Where $R$ is the area enclosed by the curves $y=1-x,y=2-x,y=1+x,y=-1+x$. Which substitution would help here? I couldn't ...
2
votes
1answer
64 views

Area Between Three Curves

Okay so I'm not just looking for answer but I really need help conceptualizing which curve when using more than 2 belongs as the "top curve". The problem I'm working with right now is $y=x^2$, ...
0
votes
0answers
10 views

Double integral (volume) of function $y^2+z^2\le a^2$ in the region $x^2+y^2\le a^2$ [duplicate]

Well, the book tells me to dray the set: $$x^2+y^2\le a^2, y^2+z^2\le a^2, (a> 0)$$ So I interpreted this as the volume of something under the region $x^2+y^2\le a^2$. This something should be ...
0
votes
2answers
53 views

Fastest way to do $\int_{-1}^1 [(1-x^2)-x^2(1-x^2)] dx$?

I'm a very lazy person and I always sum fractions wrong. I want to integrate if fast without having to expand this into a 4th degree expression: $$\int_{-1}^1 (1-x^2)-x^2(1-x^2) dx$$ what would you ...
2
votes
2answers
70 views

Other ways to compute this integral?

The following (improper) integral comes up in exercise 2.27 in Folland (see this other question): $$I = \int_0^\infty \frac{a}{e^{ax}-1} - \frac{b}{e^{bx}-1}\,dx.$$ I computed it as follows. An ...
0
votes
1answer
37 views

gaussian integral with changing prefactor

I have a problem solving this gaussian integral: $$ \int_{-\infty}^{\infty} dx\exp\left(-A(x)\cdot x{}^{2}\right) $$ While A(x)>0, which ensures that the integral doesn't diverge. I'm especialy ...
0
votes
1answer
47 views

How can $e^x$ be restated for small $x$?

Suppose I have the following equation: \begin{equation} 1=\frac{S_0}{\gamma}(1-e^{-\gamma T_n})+\lambda\int^{t_{n+1}}_{t_n}e^{-\gamma(t_{n+1}-\tau)}g(\tau)d\tau. \end{equation} If I make two ...
3
votes
1answer
46 views

Hölder type of inequality?

Is the following inequality true? I can't find a counterexample so I start to believe it is true but I do not manage to prove it :) Any ideas? Let $f$ be a compactly supported bounded twice ...
2
votes
3answers
91 views

How do I integrate $x^5(4+x^2)^{-1/2}$?

I just started learning substitution and I can't seem to solve this exercise. I'm using $x = 2sinh(t)$ Full Solution: $32\int(sinh(t)*(cosh^2(t)-1)^2=$ $32\int(sinh(t)*(cosh^4(t)-2cosh^2(t)+1)=$ ...
0
votes
0answers
11 views

Estimate a integral over a surface S with an integral over a volume V containing S in H^1

Given a function $v \in H^1(\Omega)$, a volume $ \Omega_e \subset \Omega $ and a surface $ S \subset \Omega $ contained in $\Omega_e$. Namely, $S \cap \Omega_e = S$. The statement $$ \| v ...
1
vote
5answers
88 views

Evaluate $\int\frac {\sin 4x }{\sin x}\ dx$

Evaluate $$\int \frac{\sin 4x}{\sin x} dx$$ Attempt: I've tried to use the double angle formulas and get it all into one identity, which came out as: $$ \int \left( 8\cos^3x - 4\cos x \right) dx ...
0
votes
1answer
26 views

if $f:(-1,1) \to \mathbb R$ is odd, then each indefinite integral of $f$ is even.

if $f:(-1,1) \to \mathbb R$ is odd, then each indefinite integral of $f$ is even. My attempt: Let $\int f(x) dx = F(x)+C$. $f$ is odd so $f(-x)=-f(x)$. so $\int f(-x) dx = -\int f(x) dx$ so ...
2
votes
2answers
244 views

Find the solution of the differential equation that satisfies ${{dP} \over {dt}} = 8\sqrt {Pt} ,\,P(1) = 5$

Please help. My homework is grading my answer as incorrect, but I can't tell what I did wrong. The second photo is the work of the problem done correctly but with dp/dt=2sqrt(Pt). I based my work off ...
0
votes
4answers
73 views

To show that: $\int_{-1}^{1} \frac{1}{(1+x)^{1/3}\ (1-x)^{2/3}} dx = \frac{2\pi}{\sqrt3}$

To show that: $$\int_{-1}^{1} \frac{1}{(1+x)^{1/3}\ (1-x)^{2/3}} dx = \frac{2\pi}{\sqrt3}$$...............(A) We see that $\frac{2\pi}{\sqrt3}$ can be written as $$\frac{\pi}{\sqrt3 /2} = ...
1
vote
2answers
47 views

Integration of $\int \frac{\cos2x}{\cos^2x} \ dx$

From the integral $$\int \frac{\cos2x}{\cos^2x} \ dx$$ I managed to get $$\int \frac{2\cos2x}{\cos2x+1}\ dx $$ How do I simplify it further?
0
votes
3answers
42 views

Improper integration of a general function

I am trying to find some closed form answer for the integral $$\int_0^{\infty}\frac{x^n}{(x^2+1)^n}dx,2\leq n$$ I am not sure if a closed form exists and I have been trying this integral for hours. ...
4
votes
3answers
134 views

Proof for divergence of $\int_1^\infty \cos(x^\frac{3}{4})$

As the title says, I am unable to find a proper proof for the divergence of $$\int_1^\infty \cos(x^{3/4})$$ As its is not positive, I can't use any of the divergence tests, nor compare it to a sum. ...
0
votes
2answers
42 views

$\int_{0}^{\infty }\frac{x}{(1+x^2)^n}$

Find an expression, in terms of $n$ and $a$, for $\int_{0}^{a}\frac{x}{(1+x^2)^n}dx$. For what values of $n$ does $\int_{0}^{\infty }\frac{x}{(1+x^2)^n}$ exist? State its value in terms of $n$. How to ...
1
vote
2answers
80 views

find the area of the region lying inside the circle $r=6$ and inside the cardioid $r=4-3\sin \theta$.

Well, I drew a graph to visualise it and I found the interceptions $\theta=\arcsin \left(-\frac{2}{3}\right)$. From the graph, by symmetry, I found that the area of region from $\theta$ to $\pi/2$ and ...
2
votes
1answer
68 views

Integrating $\frac{\ln{ax}}{x\ln{bx}}$

The question is just to find $$\int\frac{\ln{ax}}{x\ln{bx}}\,dx$$ With $a,b,x>0$. Now I attempted it using the substitution $u=\ln{x}$, $du=\frac{1}{x}dx$: ...
2
votes
1answer
22 views

Integrating a Gaussian over a cylinder

Consider that I am some height $r$ above a cylinder (measured from the centre) that has its axis in the direction of $z$. I wish to integrate a 2D Gaussian (of SD $\sigma$) along some part of its ...
1
vote
2answers
37 views

Direct integrals and second derivative

Let $f$ be a function continuous on $[0,1]$ and twice differentiable on $(0,1)$. Suppose that $\int_0^1 f(x)\, dx=f(0)=f(1)=0$ and $f(c)>0$ for some $c\in(0,1)$. How do i prove that there exists ...
2
votes
2answers
540 views

Prove these two integrals are equal. [closed]

Prove that: $$\forall n,m \in \Bbb N:\int_0^1 x^m(1-x)^n \,dx=\int_0^1(1-x)^mx^n \,dx$$ I really have no idea.
1
vote
0answers
60 views

Quazi-linear DE $u_{ttxx}=u_{tt}^2$

So i have a Cauchy problem based on this equation above: $$ u_{ttxx}=u_{tt}^2\\ u|_{t=0}=\phi_1(x)\\ u_t|_{t=0}=\phi_2(x)\\ u_{tt}|_{t=0}=\phi_3(x)\\ u_{ttt}|_{t=0}=\phi_4(x)$$ My idea was to ...
2
votes
1answer
69 views

How to evaluate $\int_0^A \frac{\tanh x}{x}dx$?

How to evaluate $$\int_0^A \frac{\tanh x}{x}dx$$ Where $A$ is a large positive number. The answer is: $$\ln (4e^\gamma A/\pi)$$, where $\gamma$ is Euler constant. I have no idea how to get this ...
1
vote
2answers
62 views

Compute the limit of $\int_\mathbb{R} f(x)\sin (nx)$ when $n\to\infty$, for $f \in L^1$ follow up

Let $f \in L^1(\mathbb{R})$. Find $$ \lim_{n \rightarrow \infty} \int_{-\infty}^\infty f(x)\sin(nx) dx \,. $$ Here's an attempt: If we use a simple function approximation, we end up ...
1
vote
2answers
66 views

Show that $\int_0^{\pi/2}\frac{|ab|dx}{a^{2}\cos^{2}x + b^{2}\sin^{2}x} = \frac{\pi}{2}$

I have tried out the tangent half angle substitution $\tan\frac{x}{2} = t$, which reduces the integrand to a rational expression. However, it appears to me that a more elegant solution is lurking ...
0
votes
1answer
130 views

Find the area of the region lying outside a circle r=7 and inside the cardioid r=6+7sin theta

So this is the question I have problem dealing with. I know that firstly I need to equate $7$ and $6 + 7\sin \theta$ to get the intersection. And then I am supposed to apply the formula.. But I am ...
0
votes
1answer
23 views

E[exp(-X)] when X ~ Weibull

Let $f(x) = ckx^{k-1}\exp(-cx^k)$. Can we then derive an analytical expression for the following integration? $\int_0^z f(x) \exp(-x) dx = \int_0^z ckx^{k-1}\exp(-cx^k) \exp(-x) dx$
1
vote
1answer
66 views

Evaluating $~\int_0^\pi\frac{\sin^{m-1}x}{(2+\cos x)^m}~dx$

How to find the following definite integral: $$\int_0^{\pi}\frac{\sin^{m-1}x}{(2+\cos x)^m}dx$$ I have done up to: $$\int_0^{\pi}\frac{\sin^{m-1}x}{(2+\cos ...
1
vote
1answer
57 views

Computing the area of a region from two overlapping circles

Here's my given problem: Compute the area of the region of the graph of r=4sinθ and the graph of r=4cosθ. I know from ...
0
votes
0answers
14 views

Laplace Transform Question Part B and C

Consider the following ODE for y(t): y′′+5y′+6y=e−4t+δ(t−1), y(0)=0, y′(0)=0. (a) Solve this ODE using Laplace transforms. i got this part i believe as i verified it back into the equation for ...
1
vote
1answer
21 views

Verify Stoke's Theorem with given information

Verify Stoke's theorem if $\mathbf{v} = z\mathbf{i} + x\mathbf{j} + y\mathbf{k}$ is taken over the hemispherical surface $x^2 + y^2 + z^2 = 1$ , $ z > 0$ Stoke's theorem states the following: ...
2
votes
2answers
42 views

Evaluate a certain integral over all space

Evaluate the integral $\iiint e^{-2r} \cos^2\theta \, dV $ over all space. What I have done: I wrote the limit of integration as this: $\int_0^\pi \int_0^{2\pi} \int_0^\infty r^2e^{-2r} ...
0
votes
1answer
20 views

Verifying Equivalence with Sec(x) and Identities

I'm trying to prove a couple of different problems and I'm having difficulty proving them on my own and could use a little help and advice. The first thing I needed to prove that this identity is ...
1
vote
1answer
27 views

Prove a complex integral function is holomorph and computing the derivative

I need help proving the following statement: Prove $g(z)$ = $\int_{|w-3i|=5} \frac{|w|^2}{(w-z)^4} $ is holomorph over the compliment of the circle {$w||w-3i|=5$} and computing the derivative. Here ...
3
votes
4answers
101 views

Verify ∫sec(x) =1/2 ln |(1+sin(x)) / (1-sin(x))| + C

Question says it all, how can I verify the following? $$\int\sec x\ dx=\frac12 \ln \left|\frac{1+\sin x}{1-\sin x}\right| + C$$
0
votes
0answers
33 views

Properties of the general solution?

I want find the general solution of the equation: \begin{equation} \frac{1}{2}\left(\frac{\mu_1-\mu_2}{\sigma}\right)^2p^2(1-p)^2 \frac{d^2 u}{dp^2}(p) ...
1
vote
1answer
84 views

Find the indefinite integral: $\int { {\sqrt{x+1}} \over {\sqrt{x+2} - \sqrt{x-2}} }dx$

Find the indefinite integral: $$\int { {\sqrt{x+1}} \over {\sqrt{x+2} - \sqrt{x-2}} }dx$$ I don't know how to start, multiplying by ${ {\sqrt{x+2} + \sqrt{x-2}} \over {\sqrt{x+2} + \sqrt{x-2}} }$ ...
0
votes
3answers
101 views

Integrate $\int \frac{dx}{x \ln x}\, \mathrm{d}x$ using integration by parts

I know that the integral of $\int \frac{1}{x \ln x}\, \mathrm{d}x$ can easily be obtained through substitution for $u=\ln x$ with the result of $\ln \ln x+C$. My question is if this answer (or an ...
0
votes
1answer
20 views

Question about trigonometry substitution integration.

For example, $\int \sqrt{16-9x^2}$. I know that to solve this you've to substitute $x=\frac{4}{3}\sin u$ (Through W.A). But how do I know what to let $x$ be when there is no W.A? Sorry for my english. ...
0
votes
1answer
65 views

Use fourier transform to solve second-order differential equation — an “easy” integral?

I have scoured the internet for a fully-explained solution to this problem but have found none: The problem asks to solve this differential equation for $y(t)$ using Fourier Transforms, and then ...
0
votes
0answers
31 views

How can I find $\int {\sqrt {{{\left[ 1 - {r k \cos \left( {(w - t )s + p} \right) } \right]}^2} + {{\left( {r w } \right)}^2}}} \mathrm{d}s$?

I have to integrate $$ \int {\sqrt {{{\left[ 1 - {r k \cos \left( {(w - t )s + p} \right) } \right]}^2} + {{\left( {r w } \right)}^2}}} \mathrm{d}s. $$ I have tried solving it with Mathematica, but ...
1
vote
2answers
33 views

How to find value of the given integral

Let $\Omega=\{z\in \mathbb C:Im z>0\}$ and let $C$ be the smooth curve lying in $\Omega $ with initial point $-1+2i$ and final point $1+2i$. Find the value of $\displaystyle \int ...
0
votes
1answer
25 views

inequality involving norms and integrals

For a square integrable function $f$, is the following true, and if so under what circumstances? \begin{equation} \left\Vert \int_{a}^{b}f\left(t\right)dt\right\Vert _{2}\leq\int_{a}^{b}\left\Vert ...
0
votes
3answers
78 views

Find the indefinite integral: $\int {1 \over {x^2 \sqrt {x^2-1}}}dx$ - more simple way?

Find the indefinite integral: $$\int {1 \over {x^2 \sqrt {x^2-1}}}dx$$ I solved it using Integral Substitution where $t=\arccos{1 \over x}$. But is there a more simple way? (not $x = {1 \over ...
2
votes
1answer
81 views

Double integral involving incomplete Gamma function

I want to solve an integral of the type: $\int_c^d \frac{1}{x^s} \int_{x}^{\infty} t^{r-1} e^{-t} dt dx = \int_c^d \frac{1}{x^s} \Gamma(r, x) dx$ So, the inner integral is an incomplete Gamma ...
7
votes
1answer
81 views

Evaluating an integral - is it a two dimensional beta function? This arises from a variant of Goldbach's conjecture.

Let $\gamma>0$. I would like a nice way to prove that $$\int_{\begin{array}{c} 0\leq s,t\leq1\\ s+t\leq1 ...