Question about finding the primitives of a given function, whether or not elementary.

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2
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3answers
36 views

How to prove and evaluate an Improper Integral

How to show that this improper integral converges and how to compute its value? $$ I=\int_{0}^{\frac\pi 2}\frac{\cos(2t)}{\sqrt{\sin(2t)}}\mathrm{d}t. $$ I used that the integrated function is odd so ...
0
votes
1answer
64 views

solving the integral of $e^{x^2}$

How to solve following integral? Any hints for the above integral ? $$\int{e^{x^2}} dx =?$$ I use change of variable $t=x^{2}$. so, $$\frac{1}{2}\int{\frac{e^{t}}{t^{\frac{1}{2}}}dt}$$ But I ...
1
vote
0answers
27 views

Is there a closed form expression for the following definite integrals?

I am looking for a closed form for these two integrals $$\int_{-\infty}^{-a}\text{d}x \frac{1}{|x|}e^{-\frac{1}{2}x^2\sigma^2}e^{i k |x|}+\int_a^{\infty}\text{d}x ...
0
votes
4answers
64 views

Evaluating $\int_{-a}^{a}\sqrt{a^2-x^2}dx$

Question: How to evaluate $$\int_{-a}^{a} \sqrt{a^2 - x^2} dx$$ This came up while trying to prove that the area of an ellipse is give by $\pi a b$ where $a$ and $b $are the major and ...
0
votes
3answers
58 views

Problem in indefinite integral. (Exponential)

I'm given this integral to integrate. I've no idea where to start with. Perhaps someone can give me some hints or guide me. Thanks a lot. $$\int\frac{(x^3)e^{x^2}{}}{x^2+1}dx$$
4
votes
5answers
82 views

Something wrong at $\int \frac{x^2}{x^2+2x+1}dx$

I have to calculate $$\int \frac{x^2}{x^2+2x+1}dx$$ and I obtain: $$\int \frac{x^2}{x^2+2x+1}dx=\frac{-x^2}{x+1}+2\left(x-\log\left(x+1\right)\right)$$ but I verify on wolfram and this is equal with: ...
2
votes
1answer
58 views

Arithmetic mean of $L^2$ function is $L^2$

I have found the following problem, to which I do not find the solution: Consider $f(x), x > 0$ a function such as $$ \int_0^\infty f^2(x) dx < \infty $$ and let $g(x) = \frac 1x \int_0^x ...
0
votes
0answers
25 views

Double Integral with integrand similar to bivariate normal density

I got a double integral like the following, $$\int_{0}^{\infty} \int_{y}^{\infty} xe^{-\frac{(x-by-c)^2}{2a}}ye^{-\frac{(y-e)^2}{2d}}dxdy,$$ where $a$,$b$,$c$,$d$,$e$ are viewed as some other ...
1
vote
2answers
141 views

Is this integral impossible to solve?

Is possible to express the antiderivative $$\int\frac{-3e^{-x^3}}{x^2}dx$$ in terms of elementary functions?
4
votes
2answers
54 views

Evaluate $\int \frac{dx}{(x^2 + 1)^\alpha}$

I couldn't follow a step while reading this answer. Since I do not have enough reputation to post this as a comment, I'm asking a question instead. The answer uses "partial integration" to write this ...
2
votes
3answers
175 views

Where did I go wrong when doing this integral?

This is my integral $$ \int \frac{ (2x-3)}{(x^3 +10x)}\cdot dx \\ $$ This is my work $$ \int\frac{2x}{x^3 +10x}\cdot dx-\int\frac{3}{x^3 +10x}\cdot dx\\ $$ Looking at them separately: $$ ...
5
votes
2answers
76 views

How to integrate $(e^x + 2x)^2$?

I need to integrate $\int(e^x+2x)^2dx.$ I tried breaking it into $\int(e^x+2x)(e^x+2x)dx$ and then integrating by parts, but got stuck at $$ \int (e^x + 2x)^2\,dx = ...
3
votes
1answer
68 views

Evaluating: $I_1 = \sin^{-1} \left(\sqrt{\frac{x}{x+a}}\;\right) $

$$I_1 =\int \sin^{-1} \left(\sqrt{\frac{x}{x+a}}\;\right) dx= ?$$ I tried substitution: $\sin^{-1} \left(\sqrt{\frac{x}{x+a}}\;\right) = \Xi$, but then I'm not able to do anything after the resulting ...
1
vote
1answer
27 views

calculate integral of given function

let us consider following integral while if we calculate from -infinity to plus infinity then it says that generally it should be 1/infinity +1/infinity right? which should be equal to ...
1
vote
0answers
25 views

Simulating r.v.'s from a joint density by rejection sampling in R. Continued

I wish to sample variables $v$ and $w$ from the joint density $$(v+w)e^{-\frac{(v+w)^{2}}{2x_{0}}-2\mu v-(\mu -\lambda )w},$$ where $x_0$, $\mu$ and $\lambda$ can be seen as positive constant. Since ...
0
votes
1answer
70 views

Can anyone please help with this integral. Very much appreciated..

Now , i've tried a couple of different substitutions and integrating partially but unfortunately to no luck, was wondering on your thoughts on it. I'd also be very thankful if someone were to have a ...
1
vote
0answers
41 views

Prove that these result are the same

I did this trigonometric integral in two different ways, and the results that I got were with two different trigonometric functions, $\sec x$ and $\tan x$. The integral is: $\mathbf{\int tan^{5}x \, ...
5
votes
2answers
111 views

evaluate $\int \frac{\tan x}{x^2+1}\:dx$

$$\int \frac{\tan x}{x^2+1}\:dx$$ I used By-parts method setting $u=\tan x$ and $dv=\frac{1}{x^2+1}dx$, but then I got an integral that's more complicated I also thought of trigonometric ...
1
vote
1answer
64 views

how to compute the integral $\int_0^1 (1-x^p)^n dx$?

For constants $n$ and $p$, how to compute the integral $\int_0^1 (1-x^p)^n dx$ ? I saw a solution using hypergeometric function and another using incomplete beta function here: ...
2
votes
0answers
48 views

Is this integral is right or wrong?

We did this exercise in class in a way, but at home I tried to solve it in a different way and I do not know if it is right or wrong. May you help me please? $\mathbf{\int tan^{5}x \, \, \, sec^{4}x ...
0
votes
2answers
38 views

Basic question about integrating by parts

Suppose I want to solve the following: $\int \arcsin(t) \space dt=?$ In order to solve this I would use integration by parts: $\int \ uv'dx=uv-\int \ u'v \space dx$ If I let $v'=\arcsin(t)$ then ...
3
votes
3answers
48 views

Is there another way to solve $\int \frac{x}{\sqrt{2x-1}}dx$?

$$\int \frac{x}{\sqrt{2x-1}}dx$$ Let $u=2x-1$ $du=2dx$ $$=\frac{1}{2}\int \frac{u+1}{2\sqrt{u}}du$$ $$=\frac{1}{2}\int (\frac{\sqrt{u}}{2}+\frac{1}{2\sqrt{u}})du$$ $$=\frac{1}{4}\int ...
1
vote
3answers
36 views

How to integrate $\int \frac{x^{\frac{k}{2}-1}}{1+x^k}dx$

How would I do the following integral? $$\int \frac{x^{\frac{k}{2}-1}}{1+x^k}dx$$ Where $x > 0$ and $k$ is a constant greater than $0$
0
votes
0answers
15 views

Numerical Triple integral with three other parameters in R

I am trying to integrate this function $f(u,v,w; t,x_{0},z)$ with respect to three variables, $u$, $v$, $w$, although the function also have other three parameters $t$, $x_0$, and $z$. Question: How ...
1
vote
1answer
53 views

I don't know how to tackle this integral [closed]

That is the integral: $$\int \frac{r^2}{\left(r^{2\:}+\:z^2\right)^{\frac{3}{2}}}dr$$
1
vote
1answer
35 views

variation of parameters, definite vs indefinite integrals

When performing variation of parameters to find the particular solution of a differential equation, I am confused whether the integrals should be definite or indefinite. Consider the differential ...
1
vote
2answers
63 views

Finding $\int \frac {e^x}{e^x+2}dx$

Find $\int \frac {e^x}{e^x+2}dx$ From simple division I got: $\frac {e^x}{e^x+2}=1 - \frac 2 {e^x+2}$ so we're left with finding $-\int \frac 2 {e^x+2}dx=-2\int \frac 1 {1+2/e^x}\cdot \frac 1 ...
0
votes
0answers
54 views

Is there a general way to integrate: $\int(f(x))^ndx$?

Is there a general way to integrate: $\int(f(x))^ndx$ ? For example: integrating $\int (x^2+ x^{-\frac 15})^3dx$ without expanding the parenthesis? Note: preferably indefinite integrals.
2
votes
3answers
74 views

Find the integral $\int \:x^{-\frac{1}{2}}\cdot \left(1+x^{\frac{1}{4}}\right)^{-10} dx$

Help me find the integral. I think we have to somehow replace apply. $$\int \:x^{-\frac{1}{2}}\cdot \left(1+x^{\frac{1}{4}}\right)^{-10} dx =\int \frac{1}{\sqrt{x} (1+x^{\frac{1}{4}})^{10}} dx $$
1
vote
1answer
34 views

Mixing definite and indefinite integrals

If I have the differential equation $\frac{d^2 y}{dx^2} = f(x)$ and integrate once using indefinite integrals $\frac{d y}{dx} = c_1 + \int f(x) dx $ then apply the boundary condition $\frac{d ...
1
vote
1answer
49 views

Integral including a Bessel function of the first kind

I tried to find the following integral using maple and mathematica but they would not do me the favour (only for $b=1$, but I am looking for generic real $b,a,c$). $$\int_0^a x \sin(bx)\, ...
2
votes
2answers
248 views

Solve the indefinite integral of ${\sqrt{x}\arctan\sqrt{x}\over 1+x}$

$$\int {\sqrt{x}\arctan\sqrt{x}\over 1+x}dx$$ I tried a substituition of $x=t^2$, then doing it by parts. It didn't go too well...
1
vote
2answers
244 views

Is there another simple way to solve this integral?

$$\int \frac{x(2-x^3)}{(x^3+1)^2}dx$$ Is there some simple ways to solve this integral? As my solution for this integral is very long. It's not suitable for my student.
3
votes
3answers
118 views

How to integrate $ \int \frac{x^2}{(x \sin(x)+\cos(x))^2} \mathrm{d}x$

Evaluate $$\displaystyle \int \frac{x^2}{(x \sin(x)+\cos(x))^2} \mathrm{d}x$$ Can someone just tell me the necessary manipulations? Hints will be enough. Can it be done by integration by ...
5
votes
3answers
501 views

Improper integral of a rational function!

Find the value of the integral $$\int_0^\infty \frac{x^{\frac25}}{1+x^2}dx$$ I tried the substitution $x=t^5$ to obtain $$\int_0^\infty \frac{5t^6}{1+t^{10}}dt$$ Now we can factor the denominator to ...
10
votes
6answers
971 views

Indefinite Integral with “sin” and “cos”

Indefinite Integral with sin/cos I can't find a good way to integrate: $$\int\dfrac{3\sin(x) + 2\cos(x)}{2\sin(x) + 3\cos(x)} \; dx $$
1
vote
1answer
72 views

Calculate this integral $\int \int_{R^2}x^2e^{-\sqrt{x^2+y^2}} $

So I find myself stuck on this integral where $R^2$ is the real plane: $$\int\int_{R^2}x^2e^{-\sqrt{x^2+y^2}} $$ I know I'm suposed to perform some variable substitution but I just don't see it. ...
0
votes
3answers
63 views

How to compute this integral? [closed]

I don't really know where to start with this. $$\int \sqrt{x^2+y^2+1}\quad dx$$
1
vote
4answers
70 views

Integrate:$\int({1+x-\frac{1}{x}})e^{x+\frac{1}{x}}dx$

How to find $$\int({1+x-\frac{1}{x}})e^{x+\frac{1}{x}}dx$$ I am thinking about breaking it into the form $$\int{e^x (f(x)+f'(x))}dx=e^x f(x)+C$$ Don't know how to split it. Please help.
1
vote
1answer
48 views

Closed form solution of $\int \exp(-a (b-x)^{3/2}-cx)\text dx$

Does following integral have a closed form solution (a, b, and c are constants) $$ \int \exp(-a (b-x)^{3/2}-cx)\text dx $$ If not possible, what about a function with close behavior. $$ \int \exp(-a ...
0
votes
1answer
30 views

ordinary differential equations exercise

I'd like to resolve the ode $x'(t)=\cos(\ln(1+x(t)^2))$, given the inicial value $x(0)=1$, for t belonging to the interval [0,pi]. By separated variables method I get t equal to the primitive of the ...
1
vote
0answers
85 views

A difficult integral

Well, here goes: $$\int\frac{x^2-1}{(x+1)^3\sqrt{1+3x^2+x^4}}\,dx$$ I wrote down a solution ... I am mentioning the basic steps. a) Apply the sub $x=1/t$ and get the integral into a much a better ...
3
votes
2answers
71 views

Can anyone help with these integrals?

$$1.\int{x\over \sqrt{1-\sqrt{1-x^2}}}dx$$ $$2.\int{\sqrt{x+1}+2\over (x+1)^2- \sqrt{x+1}}dx$$ $$3.\int{\sqrt[3]{1-\sqrt[4]{x}}\over \sqrt{x}}dx$$ Just need the general idea, im not expecting someone ...
2
votes
2answers
63 views

Integrate $ \int \frac { e^{\arctan(x)}}{{(1+x^2)}^{\frac{3}{2}}} \ dx $

$y=arctanx$ $tany=x$ \begin{align} \int \frac { e^{\Large\arctan(x)}}{{(1+x^2)}^{\Large\frac{3}{2}}} \ dx&=\int \frac {e^{\Large\arctan(\tan y)}}{{(1+\tan^2y)}^{\Large\frac{3}{2}}}dy\\ &=\int ...
0
votes
1answer
53 views

Integral $\int\frac{dx}{(x^3-1)^2}$

Please help. I do not know what to do. You can just show the direction where to go and I continue. Here it is: $$\int\frac{dx}{(x^3-1)^2}$$
1
vote
4answers
141 views

Help evaluating $ \int \sqrt{{x}^{2} + 3} \; dx $

Can you help me evaluating the following indefinite integral? $$ \int \sqrt{{x}^{2} + 3} \; dx $$ Please, don't give a full solution, just some hint on which method to use... ** UPDATE ** Thank ...
4
votes
2answers
67 views

Integral of $\big((1+\cos(x))\sin(x)\big)^2$

What is $$\int \big((1+\cos(x))\sin(x)\big)^2dx$$ ?
1
vote
1answer
67 views

A indefinite integral $\int 2(1+x^4)^\frac{1}{3}dx $

how can i solve this indefinite integral: $$\int 2(1+x^4)^\frac{1}{3}dx $$ I need a solution!
4
votes
2answers
56 views

Is there a notion of indefinite Lebesgue integral?

When I started studying integration rigorously via the Riemann and Lebesgue integrals, one thing that struck me is that we loose completely the concept of indefinite integrals. Integrals of functions ...
0
votes
1answer
100 views

Computing $\displaystyle \int{ \frac{\sqrt[3]{x}+2\sqrt[4]{x}}{\sqrt{x}(\sqrt{x}+\sqrt[3]{x})^2}}\,dx$

$$\int{\sqrt[3]{x}+2\sqrt[4]{x}\over \sqrt{x}(\sqrt{x}+\sqrt[3]{x})^2}dx$$ I think i need to make a substitution but am having difficulties because of the different roots. $$\int \left(\sqrt{x \over ...