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

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3
votes
3answers
735 views

Is this an elementary integral? $\int r^3\sqrt{8-r^2}dr$

In a certain integration I stumbled upon an integral which I'm not sure is simple and elementary. I wonder if it is one which is easily solvable or something which requires advanced tools to do: $$\...
2
votes
4answers
45 views

How to simplify trigonometric functions with having higher multiples of $x$ if the function is complex?

$$ \int \frac{(\cos 9x + \cos6x)}{2 \cos 5x - 1} dx $$ I know that it simplifies to $ \cos x + \cos 4x $ but I have no idea how to do that. I tried expanding $\cos 9x $ and $\cos 6x$ by using the ...
1
vote
2answers
69 views

integration of exp(cos(x-a)) dx

I would like to compute either of the following integrals: $$\int e^{\cos(x-a)} \, dx$$ or $$\int_{-\pi}^{\pi} e^{\cos(x-a)} \, dx$$ In both cases, $a$ is a constant. MATLAB doesn't seem to be ...
10
votes
2answers
638 views

A slightly problematic integral $\int{1/(x^4+1)^{1/4}} \, \mathrm{d}x$

Question. To find the integral of- $$\int{1/(x^4+1)^{1/4}} \, \mathrm{d}x$$ I have tried substituting $x^4+1$ as $t$, and as $t^4$, but it gives me an even more complex integral. Any help?
4
votes
2answers
224 views

How to integrate $\int \frac{x^{13}\ dx}{x^5 + 1}$

We get this problem from our teacher today. I only wish that it was $x^{14}$ in the numerator, so we can use substitution method: $$\int \dfrac{x^{13}\ dx}{x^5 + 1}$$ I can't find way to integrate ...
1
vote
1answer
76 views

What is the integral of $e^x a^x$

Can you confirm that my answer below is correct? $$\int (a^x e^x)dx $$ My attempt: $$\int a^x e^x \, dx = \int (ae)^x \, dx $$ $$\int a^x e^x \, dx = \frac{(ae)^x}{\ln(ae)} + C $$ $$\int a^x e^x \,...
0
votes
0answers
37 views

Integral of a square root quadratic with negative leading coefficient

I have this homework problems: $$\int \dfrac{dx}{\sqrt{-x^2 + 3x - 4}}$$ What i did was take out $\sqrt{-1}$ from denominator, and complete square. The result I get was : $$\dfrac{\ln\left|x+ \sqrt{...
1
vote
2answers
16 views

Evaluating a statement without calculating the indefinite integral

I'm cramming for a supplementary exam so you might see a ton of questions like these in the 48+ hours to come <3 The question is more of just a yes or no ; Evaluate the statement without ...
2
votes
1answer
16 views

Deriving the mean of the Gumbel Distribution

I'm trying to determine an expected value of a random variable related to the Gumbel/Extreme Value Type 1 distribution. I think the answer follows the same process as expected value of the Gumbel ...
5
votes
1answer
102 views

finding the series $\sum_{n=1}^\infty \frac{x^n}{n!} \frac{1}{n}$

My goal is to solve this series $$S(x) = \sum_{n=1}^\infty \frac{x^n}{n!} \frac{1}{n}$$ I did took the derivative first w.r.t $x$ $$S'(x) = \sum_{n=1}^\infty \frac{x^{n-1}}{n!}$$ which I ...
3
votes
1answer
190 views

help verifying equation $\int_0^ x \frac{1}{1+t^n} dt$

As a follow up to a previous posting addressing the integral of $1/ (t^n+1)$ for $n\in \Bbb{N}$ I found the following $$\int_0^ x \frac{1}{1+t^n}\, dt=\sum_{i=0}^{\infty}\frac{(i!)(n^i)x^{in+1}} {(x^...
1
vote
1answer
45 views

Integration of $\int \frac{1}{x^{1/3}(x^{1/3}-1)}dx$ [closed]

Integrate the following function $$\int \frac{1}{x^{1/3}(x^{1/3}-1)}dx$$ Could someone give me slight hint to solve this question?
0
votes
1answer
97 views

Can you solve this integral [closed]

I tried to solve this integral, but I couldn't; I don't know by what substitution or by what method. Can you help me to find its exact solution? $$\int\frac {1}{3^x+x}dx$$
0
votes
2answers
101 views

Need help solving $\int\frac{\sin(x+1)\cos(x)}{\sin^2(x)+4} dx$

As the title says I need help solving the indefinite integral $$\int\frac{\sin(x+1)\cos(x)}{\sin^2(x)+4}dx$$ Thank you for any help.
1
vote
0answers
42 views

Integrals of the form $\int x^m(a+bx^n)^Pdx$

I was reading a book on Integral Calculus, and in one chapter, the author dealt with methods of solving Integrals of the form $$\int x^m(a+bx^n)^Pdx$$ The author broke it down into 4 cases:$$$$ $...
-1
votes
1answer
57 views

Calculate $\int \frac{x^{\:}}{\sqrt{x^4+3}}\ dx$ [closed]

How to calculate $$\int \frac{x^{\:}}{\sqrt{x^4+3}}\ dx$$
1
vote
0answers
29 views

Motivation behind substitutions in an integral 1

I was reading a textbook on Integration where I came across suggested substitutions for certain types of Integrals. These were as follows: $$$$ Integrals of the form $$\int\dfrac{dx}{(ax+b)\sqrt{...
1
vote
7answers
116 views

How to integrate $\int{\frac{1}{\cos(x)}}dx$ using the substitution $u=\tan\left(\frac{x}2\right)$?

So far, I've tried out to reformulate: $$\int{\frac{1}{\cos(x)}}dx$$ to: $$\int{\frac{\sin(x)}{\cos(x)\sin(x)}}dx$$ which is basically: $$\int{\frac{\tan(x)}{\sin(x)}}dx$$ But I'm not sure if this is ...
0
votes
1answer
68 views

How do I solve the integral $\int\frac{1-\sqrt{2x+3}}{1+\sqrt{2x+3}}dx$ with help substitution? [closed]

How do I solve the integral $$\int\frac{1-\sqrt{2x+3}}{1+\sqrt{2x+3}}dx$$ with help substitution? For example, if I set $t=\sqrt{2x+3}$, would that be a possible option? And if so, how would I go ...
2
votes
1answer
108 views

Integrating $\int x^{3}e^{x}\sqrt{x\ln{x}}\,\text{d}x$

How do I integrate following integral? I have tried basic substitutions but cannot get through this. $$\int x^{3}e^{x}\sqrt{x\ln{x}}\,\text{d}x$$ Thanks.
2
votes
2answers
227 views

How to integrate $\cos^2x$? [duplicate]

It seems like I am stuck on such a simple problem: How to I find the antiderivative of $\cos^2x$? I have tried partial integration, it doesn't seem to work (for me). Some help on how to integrate it ...
1
vote
3answers
123 views

Integrating $\int\frac{x^3}{\sqrt{9-x^2}}dx$ via trig substitution

What I have done so far: Substituting $$x=3\sin(t)\Rightarrow dx=3\cos(t)dt$$ converting our integral to $$I=\int\frac{x^3}{\sqrt{9-x^2}}dx=\int \frac{27\sin^3(t) dt}{3\sqrt{\cos^2(t)}}3\cos(t)dt\\ \...
1
vote
2answers
117 views

Evaluating $\int x^2 \sqrt{1-x^2}\ dx$ [closed]

I hope I can find a way to integrate this formula without resorting to numerical techniques. \begin{equation} \int x^2 \sqrt{1-x^2}\ dx \end{equation} I am not sure if there's actually a closed form ...
0
votes
0answers
14 views

How do I find indefinite integrals of these functions?

I wanted to know the various methods to integrate functions of these type-----> $I=\int\frac{dx}{P(x)}$. Where $P(x)$ is a non-zero polynomial function. This question came when I was integrating the ...
1
vote
1answer
37 views

Solve $(x+1)dx+e^ydy=0$ at (x,y)=(0,1)

Solve at (x,y) = (0,1) $$ (x+1)dx+e^ydy=0 $$ $$ (x+1)dx=-e^ydy $$ $$ \int x*dx + \int 1*dx=-\int e^y*dy $$ $$ \frac{x^2}{2} + x + C = -e^y + K$$ For (x,y) = (0,1) $$ 1 = -e^1 + K = \frac{0^2}{2}+0+C$$...
8
votes
3answers
424 views

Is there an elegant way to evaluate $ I={ \int \sqrt[8]{\frac{x+1}{x}} \ \mathrm{d}x}$?

Is there an elegant way to evaluate the following integral? $$ I={ \int \sqrt[8]{\dfrac{x+1}{x}} \ \mathrm{d}x}$$ This seems to me a very lengthy question, yet it was given in my weekly ...
1
vote
0answers
67 views

The solution for the elliptic integral when $k>1.$ [closed]

What is the solution of the elliptic integral $\int\sqrt{1-k^2\sin^2x}dx$? Can we calculate the elliptic integral when it is definite?
2
votes
2answers
71 views

Integral of $\int{(x^2+2x)\over \sqrt{x^3+3x^2+1}} dx$

Find the integral of the following: $$\int{(x^2+2x)\over \sqrt{x^3+3x^2+1}} dx$$ Do set $u=x^3+3x^2+1$? So, $du=(3x^2+6x)dx$? And, $x^2+2x={u-1-x^2\over x}$? So then, $$\int{({u-1-x^2\over x})\...
1
vote
0answers
36 views

Difficult definite integral with a possibly real finite answer [duplicate]

I'm stumped here.The interval of integration would make me think to look for symmetry, but it's clearly not even or odd. I asked wolfram alpha to get any ideas, and it gave me a complex number as the ...
3
votes
2answers
110 views

Computing $\underset{x^2+y^2+(z-2)^2\le 1}{\int\int\int}{1\over x^2+y^2+z^2}dxdydz$ in Spherical Coordinates

Compute: $\underset{x^2+y^2+(z-2)^2\le 1}{\int\int\int}{1\over x^2+y^2+z^2}dxdydz$. Hint given: show that $\cos \theta> {r^2+3\over 4r}$ $1<r<3$ What I already did: I shift the unit ...
2
votes
1answer
61 views

Series expansion of $\int x^xdx$

The indefinite integral: $$J=\int x^xdx$$ has no known closed form solution. Expanding in series the function $f=x^x$ we get: $$f\simeq\sum_{k=0}^N \dfrac{x^k\ln(x)^k}{k!}$$ So we can write: $$J\...
1
vote
1answer
58 views

Comparison between integrals

Let's say - $F(x) > G(x)$ for all $x$ in $E$. So is it true that ? $$\int F(x)\,dx > \int G(x)\,dx$$ or I can use this statement only when the two functions are non-negative functions. ...
0
votes
0answers
36 views

how to compute $\int (\sqrt{x^3 + 1} + \sqrt[3]{x^2 + 2x} )dx $ [duplicate]

I have a problem to compute $\int_{0}^{2} (\sqrt{x^3 + 1} + \sqrt[3]{x^2 + 2x} )dx $ for $\int_{0}^{2} \sqrt{x^3 + 1} \ dx $$ \ \ = \int_{0}^{2} \sqrt{(x^{\frac{3}{2} })^{2} + 1} \ \ dx \ \ ...
2
votes
1answer
180 views

Integrating the following $\int \sqrt{\tan x+1}\,dx$

Question: Integrate the following, $$\int\sqrt{\tan x+1}\;dx.$$ Wolfram Alpha returns a non-elementary answer. Can someone please spot the mistake I have made here: First consider this integral: $...
-1
votes
2answers
61 views

Limit of Riemann Sum equal to integral [closed]

I don't understand why Limit of a Riemann sum suddenly turns into a integral. It makes no sense both intuitively and geometrically to me. In case, differentiation it is easy understand though.
8
votes
2answers
1k views

How to evaluate this monsterous integral?

How to prove this? $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac {2(x-x^2)e^{-(x^2+y^2+z^2)}}{((x-1)^2+(y-1)^2+(z-1)^2)^{3/2}}dxdydz=\frac {-4\pi e^{-3}}{3}$$ I ...
1
vote
2answers
35 views

Prove that $\int f^{−1}(x) dx = yf(y) −\int f(y) dy $

Prove that $\int f^{−1}(x) dx = yf(y) −\int f(y) dy $. Note: I came across this formula in this very interesting (at least to me) survey article on the Lambert W function: http://www.apmaths.uwo.ca/...
1
vote
3answers
69 views

How should I try to evaluate the integral $\int_a^b \sqrt{1 + \frac{x^2}{r^2 - x^2}} \; dx$

I've tried to evaluate $\displaystyle\int_{-r}^r \sqrt{1 + \frac{x^2}{r^2 - x^2}} \; dx$ on my own, but I have encountered a problem I cannot get around. The indefinite integral $\sqrt{\frac{r^2}{r^2-...
1
vote
0answers
22 views

Closed form for $\int\frac{\left((x + i) \beta\right)^\beta x^{\beta - 2}}{(x^2 + 1)^\beta} \exp\left(-\frac{\alpha}{A x}\right) \, \mathrm{d} x$

The following integral comes up in the solution of a differential equation when solved by Maple: $$ \begin{equation} \int\frac{\left((x + i) \beta\right)^\beta x^{\beta - 2}}{(x^2 + 1)^\beta} \exp\...
0
votes
1answer
30 views

Integrating by substitution containing a further factor problem

The problem: $$ {\int } (x+1)(3x+1)^9 dx $$ let u = 3 x +1 3 x = u - 1 $ x = \frac{1}{3} (u-1) $ Hence, $ x + 1 = \frac{1}{3} (u-1) + 1$ $ = \frac{1}{3} (u+2) $ This line here I do not ...
1
vote
2answers
229 views

Integrating trig function

I'm stuck at this problem: $$ \int{\sqrt{(\sin^2 x)^2 + (2\sin x \cos x)^2}dx} = \int{\sqrt{\sin^2 x \sin^2 x + 4\sin^2 x \cos^2 x} dx}$$ I tried a few trig identities: $\sin^2 x = \frac{1-\cos 2x}{...
0
votes
2answers
80 views

Solve integral $\int \frac{x+1}{x^2-2x+5} dx$

I need to solve: $$\int \frac{x+1}{x^2-2x+5} dx$$ I cann see that $D>N$ so I tried to scompose the $D$ but I get: $$x_{1,2} = \frac{2 \pm \sqrt{4-20}}{2}$$ So $\Delta < 0$ and I tried to use ...
1
vote
1answer
17 views

$\int(\int\phi(a-z)dz)dz=\Phi(a-z)$

Lets assume $\phi(a-z)$ is integrable. Can I conclude that the following integral $$\int\left(\int\phi(a-z)dz\right)dz$$ Can be expressed by a function $$\Phi(a-z).$$ So in result: $$\int\left(\...
1
vote
1answer
49 views

Find antiderivative of $ln(x)^y$ for any real y

What is this antiderivative? I have tested several values of $y$ in an online antiderivative calculator, but it's not clear how they are related. Here $y$ is fixed and I want the antiderivative with ...
2
votes
2answers
72 views

Solve the differential equation:$\frac{\,dx}{mz-ny}=\frac{\,dy}{nx-lz}=\frac{\,dz}{ly-mx}$

QUESTION: Solve the differential equation: $$\frac{\,dx}{mz-ny}=\frac{\,dy}{nx-lz}=\frac{\,dz}{ly-mx}$$ MY ATTEMPT: I tried out to proceed by using $$\frac{\,dx}{mz-ny}=\frac{\,dy}{nx-lz}=\frac{\,...
5
votes
3answers
112 views

Solution of integral $\int \frac{\sin (x)}{\sin (5x) \sin (3x)} dx$

Find the following integral $$\int \frac{\sin (x)}{\sin (5x) \sin (3x)} dx$$ I don't know how to deal with the $\sin (x)$ in the numerator. If it had been $\sin (2x)$ then we could have used $\sin (...
0
votes
1answer
47 views

Finding an antiderivative

$\newcommand{\dx}{\,\mathrm dx}$ I need to find the following: $$\int \sqrt{\frac{1-x}{1+x}} \cdot \frac 1x \dx$$ Firstly, it has to be that $x\in (-1, 0)\cup (0,1]$. From this, it is implied ...
1
vote
0answers
36 views

How to integrate $\cos2\pi\left(x+\frac{n}{x}\right)$

This is a follow up question of Integrate $\cos^2(\pi x)\cos^2(\frac{n\pi}{x})$. By using product to sum formula, this could be converted to question to integrate $$\int\cos2\pi\left(x+\frac{n}{x}\...
1
vote
1answer
58 views

Evaluate the indefinite integral $\int \frac{t\sin at}{b^2+t^2}dt$

It is known DLMF (25.2.8) that for $\Re s>0$ and for integers $N\geq 1$ $$\zeta(s)=\sum_{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-s\int_{N}^\infty \frac{x-\lfloor x \rfloor}{x^{s+1}} dx,$$ where $\...
0
votes
1answer
37 views

Proving integration formula involving the form a+bx

While trying to memorize and understand various integration formulas, I came across an integration rule stating that $$ \int \frac{1}{x^2(a+bx)^2} dx = -\frac{1}{a^2}\left[\frac{a+2bx}{x(a+bx)}+\frac{...