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

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3
votes
2answers
69 views

Evaluate $\int x \sqrt{1 - x^4} \,\mathrm{d}x$

I have the following question $$\int x \sqrt{1 - x^4} \,\mathrm{d}x$$ I know we have to use trig. substitution for this and therefore, I did the following by letting $x = \sin \theta$ and $dx = \cos ...
2
votes
2answers
38 views

Integral of $\frac{1}{x^2+1}$ using complex partial fractions.

Is there any way to evaluate the following integral via a complex partial fraction decomposition? $$ \int \dfrac{1}{x^2 + 1} \text{ d}x $$ So far I have: $$ \begin{aligned} \int \dfrac{1}{x^2 + 1} ...
0
votes
1answer
37 views

Primitive $r/(1+r^2)$ without abs()

Why should there not be an absolute value-sign instead of () when I find the primitive of $r/(1+r^2)$? Maybe it should only be there when I derive? ...
4
votes
0answers
53 views

Solving integral $\int\frac{\sin x}{1+x\cos x}dx$

How I can find the anti-derivative? $$\int\frac{\sin x}{1+x\cos x}dx$$
0
votes
1answer
63 views

Integral of $\sin|x|$

$$\int\sin|x|~dx$$ We have two cases: x less than zero, or x equals or higher than zero. $$\int_{-\infty}^0\sin(-x)~dx+\int_0^\infty\sin x~dx$$ Left side of this sum is equals to right side, so we ...
1
vote
4answers
65 views

How can I prove the integral?

Prove that $$ \int\frac{dx}{x(\log_e x)^{7/8}} = 8(\log_e x)^{1/8} $$ I am totally lost on this subject. Any help how to prove this is appreciated!
3
votes
1answer
61 views

Does this integral have any closed form? $\displaystyle\int\frac{1}{x+\sin(x+1)}\mathop{\mathrm dx}$

Does this integral have any closed form? $$\int\frac{1}{x+\sin(x+1)}\mathop{\mathrm dx}$$ I think the substitution $x=(u-1)+2\pi$ will do it, no?
0
votes
2answers
75 views

Is it possible to convert $\sigma = \int_0^\infty e^{-x^2} dx$ to an integral problem over $(0,1)$? [on hold]

Is it possible obtain a transformation to convert $\theta=\displaystyle\int_0^\infty e^{-x^2}\, dx$ to an integral problem over $(0,1)$?
3
votes
0answers
33 views

How do I integrate $\langle\nabla u,\nabla v \rangle$ in arbitrary dimensions?

I am trying to show that if $u_n$ are eigenfunctions of the Laplacian operator that make up an orthonormal basis of $L^2$, then $u_n\sqrt{\lambda_n}^{-1}$ form an orthonormal basis of $H^1_0$. I ...
2
votes
2answers
42 views

Reduction formula for $\int \frac{dx}{x^n \sqrt{ax+b}}$

I want a reduction formula for $$I_n=\int\frac{dx}{x^n \sqrt{ax+b}}$$ in terms of $I_{n-1}$. I have tried various substitutions but I just can't seem to find the right one. Any help or hints will ...
2
votes
2answers
153 views

What is the easiest way to integrate $y=\frac {x+4}{\sqrt{-x^2-2x+3}}$?

What is the easiest way to integrate $y=\frac{x+4}{\sqrt{-x^2-2x+3}}$ ? I tried to integrate it by making numerator in form: $-2x-2$ and then pulling it under differential, but the result drastically ...
7
votes
2answers
165 views

Fun Integral $ \int \frac{dx}{\cos^3 x+2\sin(2x)-5\cos x}$

$$ I\equiv \int \frac{dx}{\cos^3 x+2\sin(2x)-5\cos x}. $$ This integral does have a closed form. I am not sure where to start. We can factorize the denominator as $$ \cos^3 x+2\sin(2x)-5\cos ...
0
votes
2answers
78 views

Integral $\int\sqrt{\sin2x}\operatorname d\!x$

I tried all substitutions but failed. I need assistance to evaluate that indefinite integral. $\int\sqrt{\sin2x}\operatorname d\!x$
1
vote
1answer
24 views

Integral Confusion

Evaluate the indefinite integral: $$\int\sqrt{3-t}\;dt$$ Evaluate the indefinite integral: $$\int x^2\sqrt{x^3+9}\;dx, \quad u = x^3+9$$ I am confused on how to evaluate the two problems ...
5
votes
0answers
72 views

About the great result of $\int\underbrace{x^{x^{\cdot^{\cdot^x}}}}_m~dx$

I notice both wikipedia and mathworld have the great result of $\int\underbrace{x^{x^{\cdot^{\cdot^x}}}}_m~dx$ that: ...
0
votes
1answer
26 views

Сhange the order of integration in the double integral

I have to change the order of integration in this double integral I've decided to divide it in two similar areas D1 and D2 And I've got the following result Can You chech it and state my ...
1
vote
4answers
198 views

What is the value of $\int x^x~dx$?

I am struggling with this puzzle. Question 1. Is it possible to determine the value of the indefinite integral $\int x^x~dx$ explicitly? By "explicit" I mean without power series. Question 2. What ...
3
votes
2answers
32 views

check my solution to indefinite integral problem with arccos

So we had homework it asked us to find $$\int\arccos(x)dx$$ I have found that $$\int\arccos(x)dx=x\arccos (x)+\sqrt{1-x^2}+c$$ Is this right?
3
votes
2answers
54 views

Simple integral (involving trig)?

This seems like a simple problem, but my trig manipulations are leading to a dead end. Compute: $$\int\frac{\sin^2(x)}{1 - \tan(x)} dx$$ Working thus far: Replace $$\tan(x) = ...
0
votes
1answer
27 views

Find Indefinite of root function

I don't know how to find this strange integral $\int{\sqrt{\dfrac{x-4}{x+2}}\dfrac{dx}{x+2}}$ Please help me solve this problem
0
votes
2answers
69 views

fourier transform of sinc function

let us consider fourier transform of sinc function,as i know it is equal to rectangular function in frequency domain and i want to get it myself,i know there is a lot of material about this,but i want ...
3
votes
3answers
111 views

How do I integrate $\frac{1}{x^6+1}$

My technique so far was substitution with the intent of getting to a sum of three fractions with squares in their denominators. $t = x^2 \\ \frac{1}{x^6 + 1} = \frac{1}{t^3+1} = ...
1
vote
1answer
80 views

How to evaluate the integral : $I=\int\frac{2-x+(x-1)\ln x-\ln^2x}{(1+x\ln x)^2}dx$

How to evaluate the integral: $$I=\int\frac{2-x+(x-1)\ln x-\ln^2x}{(1+x \ln x)^2}dx.$$ Help me, thanks :/
2
votes
1answer
65 views

Integration double angle

How should i simplify this before applying integration. Have tried the $1-\cos2x=2\sin^2x$ but am still stuck on solving it $$\int\left(\dfrac{\cos2x}{1-\cos4x}\right)dx$$
6
votes
2answers
58 views

Does the following serie $\sum_{n\geq1}{\frac{1}{f(n)}}$, converge?

Let $f(x)=x$ if $0 \leq x \leq e$, $f(x)=xf(\ln(x))$ if $x>e$. Does the following serie $\sum_{n\geq1}{\frac{1}{f(n)}}$, converge ? My attempt : One can note that for all $n\geq 3$, there ...
0
votes
2answers
88 views

Integral of $\ln^2(x^2-1)/x^4$

I need to solve the following indefinite integral: $$\int \frac{\log^2(x^2-1)}{x^4}dx.$$ ($\log$ is the natural log) It's a past paper question from my uni exam so I don't think the answer is as ...
0
votes
2answers
76 views

I cannot find the following integral in an integral table.

In the appendix A of this paper there is an integral that the author says can be solved using any good integral table. However I cannot seem to find it on any integral table (ex: gradshteyn and ...
1
vote
3answers
72 views

Is the definite integral of the function necessarily the anti-derivative?

Let's say you have a function defined as $$g(x)=\int_1^xf(t)dt$$ By the integral definition, g(x) is the area under the curve of f(x) from 1 to x. eg: g(5) is the area under f(x) from 1 to 5. I ...
2
votes
2answers
50 views

Integrate $\int\frac{5x-7}{x^2-3x+2}$

I want to integrate $\int\frac{5x-7}{x^2-3x+2}$ but my result differs from the one on Wolframalpha http://www.wolframalpha.com/input/?i=integrate+%285x-7%29%2F%28x%5E2-3x%2B2%29 I did the following ...
0
votes
1answer
43 views

Indefinite Integral Question - What kind of substitution?

I've been trying to solve this integral for the past two hours, but haven't gotten anywhere: $$ \int \frac {dx}{2\sqrt{x-4}+x} $$ I've tried various kinds of substitutions to no avail. Even just ...
2
votes
1answer
25 views

Problem understanding integral evaluation

I am having trouble understanding the evaluation of an integral. Do we just separate the integrals and evaluate them? Is it like normal integration? I have provided an example below taken from one of ...
4
votes
2answers
96 views

Evaluate $\int\frac{\sqrt {25 - x^2}}{ x^4}$

I'm pretty sure the method used is trig substitution. But I'm having trouble setting up and solving the problem.
-3
votes
3answers
67 views

If integrating $y$, do I put $dy$ or $dx$ in the notation? [closed]

Simple question. When taking the integrate of $y '$ (we're not talking about an actual function, just the letter "$y '$" so I can show that integration and derivative cancel each other out) do I add ...
3
votes
1answer
39 views

Integrate irrational function

I need to solve an indefinite integral $$ I=\int\sqrt{x^4+2x^2-1}\,x\,dx. $$ Substituting $x^2=t$ yields $$ ...
3
votes
4answers
90 views

Evaluate the integral $(x-2) e^x$

I think this problem can be solved using integration by parts. So I set it up as $u = e^x$ and $du = e^x, dv = (x-2)$ and x = (x^2)/(2)-(2x) But I don't think I'm getting the right answer. Are my $u$ ...
4
votes
2answers
140 views

How to integrate this integral

I am having trouble solving this (note: we have not studied it yet nor was Google of any help) $$\int e^{x^x}\, dx$$
5
votes
6answers
303 views

How to integrate $\displaystyle 1-e^{-1/x^2}$?

How to integrate $\displaystyle 1-e^{-1/x^2}$ ? as hint is given: $\displaystyle\int_{\mathbb R}e^{-x^2/2}=\sqrt{2\pi}$ If i substitute $u=\dfrac{1}{x}$, it doesn't bring anything: ...
1
vote
3answers
100 views

Help with integral $\int\frac{1}{\sqrt{\tan x}}dx$

I tried to solve by parts but it did not help.
1
vote
2answers
47 views

Help with integral with $\arcsin x$.

$$\int \frac{(1+x^2)\arcsin x}{x^2\sqrt{1-x^2}}dx$$ I saw that $$(\arcsin x)'=\frac{1}{\sqrt{1-x^2}}$$ and I tried to solve it "by parts"
0
votes
3answers
99 views

Evaluating $\int \frac{\operatorname d \! x}{\sin^4{x}+\cos^4{x}+\sin^2{x}\cos^2{x}}$

How do you integrate $$\frac{1}{\sin^4{x}+\cos^4{x}+\sin^2{x}\cos^2{x}}$$ or simply $$\frac{1}{1-\left(\frac{\sin{2x}}{2}\right)^2}.$$
2
votes
2answers
107 views

evaluation of $\int \cos (2x)\cdot \ln \left(\frac{\cos x+\sin x}{\cos x-\sin x}\right)dx$

Evaluation of $\displaystyle \int \cos (2x)\cdot \ln \left(\frac{\cos x+\sin x}{\cos x-\sin x}\right)dx$ $\bf{My\; Try::}$ First we will convert $\displaystyle \frac{\cos x+\sin x}{\cos x-\sin x} = ...
2
votes
1answer
87 views

How would I integrate $e^{e^x}$?

Is there a way to integrate: $e^{e^x}$ without using a Taylor or McLaurin Series expansion?
2
votes
3answers
80 views

Find the integral : $\int\frac{dx}{x^\frac{1}{2}+x^\frac{1}{3}}$

Find the integral : $\int\dfrac{dx}{x^\frac{1}{2}+x^\frac{1}{3}}$ Please guide which substitution fits in this I am not getting any clue on this .. thanks..
4
votes
2answers
130 views

Help solving an integral.

$$\int \frac{\sqrt{t+2}}{e^t}\,dt$$ I have tried integration by parts, but that is leading me no where. I typed it into Wolfram Alpha, but don't know much about erf function, just know what ...
0
votes
1answer
13 views

I need to find know how to integrate $x$ multiplied by a function to a power that is a fraction.

I know how to find integral functions normally, but when I try to find it from say $x\sqrt{4-x^2}$, I get completely lost.This screws me up in both indefinite and definite integration, so please help
-1
votes
0answers
66 views

difficult trigonometry integral

I need to solve this integral involving trigonometry: i managed to get to this level: while t is tan(x/2) Thanks in advance.
1
vote
1answer
57 views

Evaluating $\int \cos^{-1}\left(\frac{x^2+a^2}{x^2-b^2}\right)x^2 dx$

How to evaluate the integral $$\int \cos^{-1}\left(\frac{x^2+a^2}{x^2-b^2}\right)x^2 dx$$$a<b.$ I posted a similar question here. Thanks in advance.
3
votes
3answers
99 views

Series Expansion Of An Integral.

I want to find the first 6 terms for the series expansion of this integral: $$\int x^x~dx$$ My idea was to let: $$x^x=e^{x\ln x}$$ From that we have: $$\int e^{x\ln x}~dx$$ The series expansion of ...
-2
votes
3answers
88 views

Evalute This Integral Function Using Trigonometric Function?

Evaluate$$\int x\sqrt{x^2 - 4}\,dx$$using trigonometric functions.
7
votes
4answers
758 views

Why should we get rid of indefinite integration?

It is the very symbol of "indefinite integral" that is flawed and confusing. It should be removed and kept only as a "guilt practice", like treating $dy/dx$ as a real fraction and things like that. ...