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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-2
votes
1answer
30 views

Evaluating an integral with unspecified functions $f,g$, given other integrals with these functions

Suppose that $$\int_6^8(3f(x)-x)\,\mathrm dx=6$$ and $$\int_8^6(2x+4g(x))\,\mathrm dx=-8$$ Evaluate $$\int_8^6 (f(x)-5g(x))\,\mathrm dx$$ I have a problem. So, this one question asks me ...
1
vote
0answers
15 views

What assumptions should be made?

take a problem like A trough is 12 feet long and 3 feet across. Its ends are isosceles triangles with altitudes of 3 feet. Water is being pumped into the trough at 2 cubic feet per minute. How fast ...
1
vote
0answers
46 views

Prove the given two integrals are not equal

I am stuck with following problem: Prove the following two integrals are not equal: $$ \int_{-\infty}^{\infty} p(y-c)\log \big(p(y-c)+p(y+c)\big)dy \neq \int_{-\infty}^{\infty} p(y+c)\log ...
2
votes
1answer
50 views

If $\int \dfrac{f(x)}{x^2(x+1)^3}\hspace{1mm}dx$ is a rational function, and $f$ is quadratic function, such that $f(0)=1$. Then Find $f'(0)$

If $\int \dfrac{f(x)}{x^2(x+1)^3}\hspace{1mm}dx$ is a rational function, and $f$ is quadratic function, such that $f(0)=1$. Then Find $f'(0)$ This looks like an interesting problem with an elegant ...
0
votes
1answer
30 views

Proving an integration with a modified Bessel function and an exponential

I am trying to prove the following identity: where $\mu, h, H$, and $\tilde{\gamma}$ are real constants. The only hint that I have is use the relation between the modified bessel function of the ...
0
votes
0answers
55 views

How can I evaluate this integral?? [duplicate]

integral $\int_{0}^{\infty} \frac{cosx}{x^2+1} dx$? I got the answer is $\frac{\pi}{2e}$ by using Wolfram. But can't do it by myself... need some help
0
votes
0answers
29 views

Bochner vs. Lebesgue

I'm trying to prove that for complex functions $f:\Omega\to\mathbb{C}$ that are not a priori measurable that: $$f\text{ Bochner integrable}\iff f\text{ Lebesgue integrable}$$ Basically it reduces to ...
0
votes
4answers
67 views

What is the most efficient way to integrate $(x-3)\sqrt{x^2+3x-18}$?

I can do the problem, but it is becoming so big,that I do not feel to do it anymore. Can anyone give the shortest method for this problem? $$\int (x-3)\sqrt{x^2+3x-18}\,dx $$
0
votes
1answer
34 views

how to remove modulus signs after integrating

$$ \frac{dy}{dt} + k\frac{t^2 -3t + 2}{t+1}y = 0,\ \ \ \ \ \ \ y(t_0=0)=A>0\\ -\int \frac{k}{y} dy = \int (t-4 + \frac{6}{t+1}) dx $$ After integrating the above how do you express $y$ in terms ...
4
votes
1answer
55 views

Integrate $\int\sqrt\frac{\sin(x-a)}{\sin(x+a)}dx$

Integrate $$I=\int\sqrt\frac{\sin(x-a)}{\sin(x+a)}dx$$ Let $$\begin{align}u^2=\frac{\sin(x-a)}{\sin(x+a)}\implies ...
3
votes
2answers
30 views

Existence and uniqueness of weights for the rule $\int_a^b f(x) \ = \ \sum_{0 \leq k \leq n} w_k f(x_k)$

I want to establish this statement: If $a<b$ and $\{x_0,x_1, \cdots x_n\} \subset \mathbb{R}$ distinct, then there is one and only one set of weights $\{w_0, \cdots w_n \} $ such that $\int_a^b ...
1
vote
3answers
36 views

integrate $\int e^{-iwt}dt$

I have this integral: $$ \int e^{-iwt}dt$$ I know that $\int e^{kx}=\frac{e^{kx}}{k}$ so therefore the $ \int e^{-iwt}dt$ would be $\frac{e^{-iwt}}{-iw}$ but Wolfram Alpha says that it is $\int ...
7
votes
0answers
105 views

Evaluating $\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$

What tools would you recommend me for evaluating this integral? $$\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$$ My first thought was to use beta function, but it's hard to get ...
0
votes
2answers
48 views

Can anybody prove why integral of f*f from 0 to 1 not 0? [on hold]

If I have a function f, which can be all real polynomials, Why integral of f * f on [0,1] is not equal to 0 ? I know intuitively, but I need to see the proof
0
votes
2answers
33 views

Integrating this improper integral to test for convergence?

I'm trying to integrate this: $$\int^\infty_0 \frac{8}{\sqrt{e^{x}-x}} \,dx$$ And use the Direct Comparison Test to find out whether it diverges or converges. I looked at a similar problem: and I ...
0
votes
1answer
79 views

Finding total work by integration

The following tank is completely filled with water. Find the total amount of work done in pumping water out of the outlet. Note that the density of water is 1000 kg/m$^3$ I feel like I am ...
0
votes
0answers
10 views

Integral formulation for LDE

I am trying to put the system in a integral formulation. All goes well for the first integration as I obtain What I don't know is how to perform the second integration in this last term. My ...
1
vote
2answers
61 views

Indefenite Integral requiring substitution

Can someone please help me find a useful substitution for the following integral: $$\int \frac{1}{\sqrt{x}(1+\sqrt{x})^2}dx$$ I tried letting $ u = \sqrt{x} $ But I couldn't proceed. Please help.
0
votes
2answers
64 views

Can I integrate $\frac{x}{1-x}$ by substitution?

I saw a person use substitution like this: $$\int \frac{x}{1-x} dx$$ Let $u= (1-x)$, $x= 1-u, du= -1\cdot dx$ $\Rightarrow$ $-du=dx$ $$\int \frac{1-u}u (-du)$$ Can I use substitution like this? I ...
3
votes
1answer
51 views

Another parametric integral relating to hyperbolic function

if $0<a\leq1$, then canwe get a closed form of $$I(a)=\int_0^\infty\frac{x}{\tanh x}\frac{1}{\cosh^2(ax)}dx.$$ In fact,if $a=1$,$I(a=1)=\pi^2/8$.
4
votes
2answers
50 views

parametric integral relating to hyperbolic function

Suppose that $a$ is real number such that $0<a<1$, how can we calculate $$ I(a)=\int_0^\infty \big(1-\frac{\tanh ax}{\tanh x}\big)dx .$$ As for some speical cases, I can work out $I(1/2)=1$. ...
0
votes
2answers
109 views

How to find $P(X>x)$ when the density is known but the integral does not seem to converge

I am trying to evaluate $$P(X>x) = \int_x^{\infty } t^{\kappa } \exp{\left(-\rho t^{\alpha\kappa + 1}\right)} \, dt$$ where $\kappa$, $\rho$ and $\alpha$ are all constants. I have tried some ...
-4
votes
3answers
38 views

This is the question about integration. I want to know how to approach this question. [duplicate]

My solution makes same loop, which eventually makes the equation as 0 = 0 form.
0
votes
1answer
36 views

Integration by Parts and Convergent/Divergent Series Calculus

We are asked to integrate: $$\int x (lnx) dx$$ Integration by parts gives us: (using L-I-A-T-E) $$u = lnx$$ $$ du = (1/x)dx$$ $$ dv = xdx $$ We find v by integrating dv: $$ v = (1/2)x^2 $$ ...
3
votes
1answer
44 views

Integral of [(1+2y^2)/(3-y)]dy (obtained from a differential equation)

This question actually arises from this Differential Equations question: Find the family of solutions for: $\displaystyle(1+2y^2)\frac{dy}{dx} + (3-y)\cos x = 0$ I ruled out the methods I've so far ...
1
vote
2answers
46 views

Integration of $1/(x^2+x\sqrt{x})$

The question is to evaluate $\displaystyle7\int\frac{dx}{x^2+x\sqrt{x}}$. My solution is attached. The problem of my solution is if I use partial fraction, loop will be made, and this makes ...
-2
votes
2answers
46 views

Integration of $(5x^2+2x-5)/(x^3-x)$

The problem is to evaluate $\int \frac{5x^2+2x-5}{x^3-x}\,dx$. This is the solution that I tried: I really have no idea of this problem. After check my solution, if there are any problem that ...
0
votes
2answers
54 views

This is the question about integration.

My idea is to use substitute integration. Since there is square root of (1-x^2), I made x = cos^2t, and then eliminated square root. I don't know why my answer is wrong. I already conducted ...
6
votes
1answer
127 views

Closed form for $\int_1^\infty\frac{\operatorname dx}{\operatorname \Gamma(x)}$

Is a closed form for $$\int\limits_1^{+\infty}\frac{\operatorname dx}{\operatorname \Gamma(x)}$$known? I tried to find it, but all well-known integrals involving gamma-function (such as of ...
0
votes
6answers
85 views

For polynomials $f,g$, why is $\int_0^\infty \frac{fg}{e^x}\, dx$ absolutely convergent?

Why does the integral $\displaystyle \int_0^\infty \frac{fg}{e^x}\, dx$ have to be convergent for all real polynomials $f$ and $g$? Can anybody give me a proof?
0
votes
2answers
47 views

What is the solution of $\int \frac{a}{a^2+x^2}dx$?

What is the solution of $\int \frac{a}{a^2+x^2}dx$? I have tried: $\frac{a}{a^2+x^2}=\frac{\frac{1}{a}}{1+(x/a)^2}=\frac{1}{a} \frac{1}{1+(x/a)^2}$ by multiplying both parts of the fraction by ...
-1
votes
0answers
36 views

$\int \left(\int_a^x f(t)\, dt\right)\, dx$ [on hold]

For a continuous function $f(x)$ has anyone thought about $$\int \left(\int_a^x f(t)\, dt\right)\, dx$$ That is, what is the integral of an area function. These (area functions) are commonly brought ...
1
vote
2answers
77 views

What do we mean by derivative of a function? What does it tell? [duplicate]

Taking the derivative of any kind of function is easy but I don't know why we take the derivative? Like $f(x)=x^2$ has the derivative $2x$, so what does it mean? I don't know how to define ...
-3
votes
2answers
67 views

Could someone show me the steps in finding this integral? [on hold]

The integral is: (I have no idea how to use integration symbols, bear with me: $$ \int_{-\infty}^{2} 0.1 \ e^{-0.2 |x|} \;\mathrm{d}x$$ I need help ASAP. Please assist. Thank-you!
2
votes
3answers
42 views

How find this integral $\iint_{D}(x^2y+xy^2+2x+2y^2)dxdy$

let $$D=\{(x,y)|y\ge x^3,y\le 1,x\ge -1\}$$ Find the integral $$I=\dfrac{1}{2}\iint_{D}(x^2y+xy^2+2x+2y^2)dxdy$$ My idea: ...
7
votes
2answers
141 views
+100

Closed form of $I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $

I'm looking for a closed form of this integral. $$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$ where $\operatorname{Li}_2$ is the dilogarithm function. A numerical ...
0
votes
5answers
90 views

How to prove that the function $f(x)=0.1\,e^{-0.2|x|} $ is a probability density, and then use it?

So here's the integral, I'm having a hard time solving it. I even tried integration software, but it didn't help: $$ I=\int_{-\infty}^{+\infty}f(x)\,dx,\qquad f(x)=0.1\,e^{-0.2|x|} $$ The question ...
1
vote
2answers
58 views

An Elliptic Integral - What's the Simplest Answer?

I have $$ \int_{0}^{2\pi}d\theta\left(R^{2}-\epsilon^{2}\right)\sqrt{R^{2}-\epsilon^{2}\sin^{2}\left(\theta\right)} $$ which Mathematica thinks is $$ ...
0
votes
1answer
25 views

Set up triple integral for volume (cylindrical coordinates)

I am given the following question Let $D$ be the region in $\mathbb{R}^3$ that lies within $x^2 + y^2 =4$, underneath the surface $z= 4- x^2 - y^2$ and above the surface $z=- \sqrt{9-x^2 - y^2}$ ...
2
votes
3answers
50 views

Finding the indefinite integral of a root function

I'm stuck on a particular integral problem. The problem is stated as: $$\int x \sqrt{2x+1} dx$$ My working thus far: $$\int x \sqrt{2x+1} dx = \frac{1}{2}x^2\frac{2}{3}(2x+1)^\frac{3}{2}$$ ...
1
vote
2answers
35 views

Finding the Limits of the Triple Integral (Spherical Coordinates)

Let $D$ be the region in $\mathbb{R}^3$ below $z=-\sqrt{x^2 + y^2}$ and above $z=-\sqrt{4-x^2 -y^2}$. Rewrite \begin{align*}\iiint \limits_D z^2 dV\end{align*} using Spherical Coordinates. I ...
0
votes
1answer
45 views

This is question about integration. I want you to check error.

Please tell me which part is wrong, or if there are better solution, please let me know. Thanks.
1
vote
1answer
29 views

Why $f (x):= \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)}$ only belongs to $L^2(0, \infty)$

This is a result given in Royden and Fitzpatrick (p. 143). Show that $$ \int_0^\infty \left[ \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)} \right]^p < \infty $$ if and only if $p=2$. That ...
4
votes
0answers
100 views
+200

Closed form for integral $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $

I'm looking for a closed form of this definite iterated integral. $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $$ From Vladimir ...
1
vote
1answer
23 views

Evaluate an integral involving tangent and secant. [on hold]

![I tried several methods that I could do. First method was I changed tan^2x = sec^2x-1, and then substitute secx to t, but it doesn't work. Second method was to use substitute tan^2x = v, secx = ...
1
vote
0answers
30 views

Integral equation solution

I have an integral equations of the form $ \int s R(s) =s f(s)-\int f(s)ds \tag 1$ Can we solve this integral equation for $f(s)$ interms of $s,R(s)$ ? Means $R(s)=\psi(s,R(s))$ (with out integral ...
-1
votes
1answer
22 views

Integration of exponential with a complex [on hold]

i want to prove the left side of the equation to the right side, can some one please help me with this
0
votes
0answers
52 views

How to find if an integral is possible to compute: Failing to solve integral for quadratic functional

I am trying to solve the below integral, and no computational method seems to be capable of solving this, nor can I do it by hand. Any ideas? $$\int_{t_0}^{t_1}[a(t)((2\dot{x^*}\dot{\eta} + ...
1
vote
0answers
45 views

Looking for advice with the following integral

I have the following integral to evaluate: $$\frac{1}{f(t)} \int_0^t s^m f(s) \sin(ps) \mathrm{d}s \quad m,p \in \mathbb{R}$$ I'm unable to proceed with this integral as it is non-trivial. Even using ...
0
votes
1answer
20 views

Vitali Set: Inner Measure vs. Outer Measure

Context Nonlinearity in general of the Lebesgue integral for nonmeasurable functions reduces in some sense to inner and outer measure of nonmeasurable sets: ...