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

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4
votes
5answers
74 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
53 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
21 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
136 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
53 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
67 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
63 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
13 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
500 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
968 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
66 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 ...
0
votes
1answer
39 views

Is there another way than linearization?

$$I= \int {\sin^mx \cos^nx }dx$$ I need a Hint on doing this integral a Successive Partial Integration but it seems that the problem shows up when $ m = 2k $ and $ n = 2p$ where $p,m \in \mathbb{N}$. ...
2
votes
4answers
100 views

Integration by parts, What I should do next

I'm integrating a function by parts but I get stuck in a part. Since here, What should I do next?
0
votes
2answers
38 views

Indefinite Integral:Can anyone kindly give an appropriate technique?

Indefinite Integral:I tried this integral by substitution,but cannot find any appropriate derivative. I tried it by product rule but ended in a complex form. Can anyone kindly give an appropriate ...
1
vote
5answers
61 views

Evaluate $\int x^3(x^2+7)\ dx$

I'm trying to find the indefinite integral of $$\int x^3(x^2+7)\ dx$$ and I've seem to have forgotten how to do it in this case. So if anyone can refresh my memory, I'd appreciate it.
5
votes
1answer
34 views

Integration of $\int \frac{\arcsin{e^x}}{e^x}dx$

I've got a problem with this integral: $$\int \frac{\arcsin{e^x}}{e^x}dx$$ I got such a result: $$\int\frac{\arcsin{e^x}}{e^x}dx=-\frac{\arcsin{e^x}}{e^x}-\ln|\sqrt{e^{-2x}-1}+e^{-x}|+C$$ but the ...