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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3
votes
1answer
33 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 ...
2
votes
2answers
21 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 ...
0
votes
3answers
30 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 ...
2
votes
0answers
47 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
42 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
29 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
29 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 kg/m$^3$ I feel like I am headed ...
0
votes
0answers
7 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
56 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
61 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
45 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
49 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
63 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
34 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: $$ v = (1/2)x^2 $$ ...
1
vote
1answer
30 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: (1+2y^2)(dy/dx) + (3-y)cosx = 0 I ruled out the methods I've so far learned in class ...
1
vote
2answers
45 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
44 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
52 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 ...
4
votes
0answers
61 views

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

Is a closed form for $$\int\limits_1^{+\infty}\frac{dx}{\Gamma(x)}$$known? I tried to find it, but all well-known integrals involving gamma-function (such as of $\log\Gamma(x)$ or the like) don't ...
0
votes
6answers
73 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
3answers
43 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
35 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
75 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
64 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
39 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: ...
4
votes
2answers
86 views

Closed form of $\int_0^1 \frac{\operatorname{Li}_2\left( \sqrt{t} \right)}{2 \, \sqrt{t} \, \sqrt{1-t}} \,dt $

I'm looking for a closed form of this integral. $$I = \int_0^1 \frac{\operatorname{Li}_2\left( \sqrt{t} \right)}{2 \, \sqrt{t} \, \sqrt{1-t}} \,dt ,$$ where $\operatorname{Li}_2$ is the dilogarithm ...
0
votes
5answers
86 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
56 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
24 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
48 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
30 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
27 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
69 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
24 views

Looking for advice with the following integral

I have the following integral to evaluate: $$ \frac{1}{f(t)}\int_0^t t^m (t + n)^o \sin(pt) \mathrm{d}t \quad m,n,o,p \in \mathbb{R}$$ I'm unable to proceed with this integral as it is non-trivial. ...
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: ...
1
vote
1answer
62 views

Numerical value of $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $

Could somebody give me a numerical value for this integral? $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $$
2
votes
2answers
67 views

computing integral without softwares: $\int \frac{2x+3}{x^2+\sqrt{1-x^2}}dx$

I was wondering if this integral can be solve without wolfram and others: $\int \frac{2x+3}{x^2+\sqrt{1-x^2}}dx$ Thanks.
6
votes
2answers
105 views

Computing in closed form $\sum_{n=1}^{\infty}\frac{\operatorname{Ci}\left(\frac{3}{4}\zeta(2) \space n\right)}{n^2}$

What tools would you recommend me for computing the series below? $$\sum_{n=1}^{\infty}\frac{\operatorname{\displaystyle Ci\left(\frac{3}{4}\zeta(2) \space n\right)}}{n^2}$$ I lack the starting ...
2
votes
3answers
67 views

How to integrate $\frac{y^2-x^2}{(y^2+x^2)^2}$ with respect to $y$?

In dealing with the integration, $$\int\frac{y^2-x^2}{(y^2+x^2)^2}dy$$ I have tried to transform it to polar form, which yields $$\int\frac{\sin^2\theta-\cos^2\theta}{r^2}d(r\cos\theta)$$ But, what ...
1
vote
1answer
44 views

Integration by parts, proving inductive case

${1\over2}\int_{-\pi/2}^{\pi/2}cos^{2n-1}(x) dx$ Inductive step: Show that the $integral={(2n-2)(2n-4)...\over (2n-1)(2n-3)...}$ for $n\ge2$ $T(n+1)$=... Attempted int. by parts using ...
2
votes
2answers
47 views

Where should I place the notorious '+c'?

Consider the following proof - $$I=\int \sin (\ln x)dx\\\iff I=\sin(\ln x)x-\int\frac{ \cos (\ln x) }{x}\cdot {x} dx \\\iff I=x\sin (\ln x)-\int\cos(\ln x)dx\\\iff I=x\sin(\ln x )-[x\cos(\ln ...
1
vote
3answers
38 views

finding an indefinite integral of a fraction

(a) Show that $\frac{4-3x}{(x+2)(x^2+1)}$ can be written in the form ${\frac{A}{x+2} + \frac{1-Bx}{x^2+1}}$ and find the constants $A$ and $B$. (b) Hence find ...
0
votes
2answers
47 views

Easy question on integrals

I have some problems understanding this inequality: $$\int_{x-\varepsilon x}^x \vartheta\left(t\right)dt \leq \vartheta\left(x\right)x\varepsilon$$ where $\vartheta\left(x\right)$ is the Čebyšëv (or ...
4
votes
1answer
82 views

Stuck on this intergral $\int^\frac{\pi}{3}_\frac{\pi}{4} \frac{\tan^2x}{x-\tan x} dx $ calculus I

$$\int^{\pi/3}_{\pi/4} \frac{\tan^2x}{x-\tan x} dx $$ this is that I have tried $$\int^{\pi/3}_{\pi/4} \frac{\frac{\sin^2x}{\cos^2 x}}{x-\frac{\sin x}{\cos x}} dx $$ $$\int^{\pi/3}_{\pi/4} ...