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

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2
votes
5answers
163 views

Evaluating $\int \sqrt{x^2-3}\:dx$

I need to solve: $$\int \sqrt{x^2-3} \, dx.$$ So I use the substitution: $$x=\frac{\sqrt 3}{\cos(t)}$$ $$dx= \frac{\sqrt 3 \sin(t) \, dt}{\cos^2(t)} $$ and I get $$3\int \frac{\sqrt{\frac1 {\...
1
vote
2answers
99 views

A formula for $\int\limits_0^\infty (\frac{x}{e^x-1})^n dx$

The Stirling numbers of the first kind $\begin{bmatrix} n \\ k \end{bmatrix}$ are defined by $\sum\limits_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix}x^k:=\prod\limits_{k=0}^{n-1}(x+k)$ with $n\in\...
1
vote
3answers
88 views

Is it possible to $\int \sqrt{\cot x}$ by hand

$$\int \sqrt{\cot x}{dx}$$ $$\int \sqrt{\frac{\cos x}{\sin x}}{dx} $$ Using half angle formula $$\int \sqrt{\frac{1-\tan^2 \frac{x}{2}}{2\tan \frac{x}{2}}}{dx}$$ But I am not getting any lead from ...
2
votes
2answers
89 views

How to evaluate $\int\frac{dx}{(2\sin x+\sec x)^4}$?

I tried a lot but I am not able to get a start. Can anyone give me the start of this question $$ \int\frac{dx}{(2\sin x+\sec x)^4} \ ? $$
1
vote
1answer
34 views

Why do I get two answer when calculating this integral from two ways?

Assuming $a(t)=a_0\sin(\omega t)$, $v(0)=0$ and $x(0)=0$. I hope you know about basic relation between position, velocity and acceleration. They are derivatives of the proceeding one. I went on ...
0
votes
2answers
37 views

Power Rule for Indefinite Integrals

To prove $\int x^p \, dx = \frac{x^{p+1}}{p+1} + C$, my calculus textbook writes: $$F '(x) = \frac{d}{dx} \left(\frac{x^{p+1}}{p+1} +C\right) = \frac{d}{dx} \left(\frac{x^{p+1}}{p+1}\right)+\frac{d}{...
7
votes
1answer
87 views

$\int \frac{dx}{\tan x + \cot x + \csc x + \sec x}$

$$\int \frac{dx}{\tan x + \cot x + \csc x + \sec x}$$ $$\tan x + \cot x + \csc x + \sec x=\frac{\sin x + 1}{\cos x} +\frac{\cos x + 1}{\sin x} $$ $$= \frac{\sin x +\cos x +1}{\sin x \cos x}$$ $$t= \...
3
votes
1answer
75 views

What approach should I use to solve integrals like this?

$$\int{\sqrt{1-{x^3}}}dx$$ I tried with $t=x^3$ but then I have the $3x^2$ dt that I can't get rid of.
0
votes
1answer
30 views

Relation of not solvable indefinite Integrals to the Galois theory?

Is there any simple explanation around why the following indefinite integral has not any solution? Is it related to the Galois theory? If yes, How? $$ \int\frac{1}{1+e^{-x}} dx = Li_2(e^{-x})+x\log(e^...
4
votes
4answers
62 views

How to solve $\int \frac{1}{x^2+4x+7} dx$?

How to solve $\int \frac{1}{x^2+4x+7} dx$? I think the first step is to write it in the following form: $$\int \frac{1}{(x+2)^2+3} dx$$
0
votes
1answer
46 views

solution of the ODE $u du =ydx+xdy$

In this case $u=u(x,y)$. When I saw this I just went on to taking iindefinite integral both sides yielding $ u^2=4xy+K $. Yet, the book I am using now got $udu=d(xy)$, which yields $ u^2=2xy+K$. I'm I ...
-2
votes
0answers
31 views

Find the primitive functions in given intervals [on hold]

I can easy compute the indefinite integral, but I have a problem with understanding the condition and I don't know what to do next - I don't know how to find a primitive function in a given interval. ...
3
votes
1answer
69 views

Another way to evaluate $\int\frac{\cos5x+\cos4x}{1-2\cos3x}{dx}$?

What I've done is this:$$\int\dfrac{\cos5x+\cos4x}{1-2\cos3x}{dx}$$ $$\int \dfrac{\sin 3x}{\sin 3x}\left[\dfrac{\cos5x+\cos4x}{1-2\cos3x}\right]{dx}$$ $$\dfrac {1}{2}\int\dfrac{\sin 8x -\sin 2x +\sin ...
0
votes
2answers
100 views

Another (perhaps tricky) integral.

While solving my Math paper, I came across this integral, and I can't see any way to solve it. At least, any easy way. The integral is- $$ \int{x^{2} \over 1 + x^{5}}\,\mathrm{d}x $$ I'm not even ...
0
votes
3answers
74 views

How to integrate $\int \frac{3+2\cos x}{(2+3\cos x)^2}{dx}$ via substitution?

$$\int \frac{3+2 \cos x}{(2+3 \cos x)^2}{dx}$$ $$\int \frac{2}{3(2+3 \cos x)}{dx} +\int \frac{5}{3(2+3 \cos x)^2}{dx}$$ I can't think of better substitution . Please tell me what will be better ...
3
votes
1answer
99 views

Integrate the following: $\int \frac {\log x}{\sqrt {1-x^2}}dx$

What I tried : $\int \frac {\log x}{\sqrt {1-x^2}} dx$ = $\log x \int \frac {1}{\sqrt {1-x^2}} dx$ - $\int \frac{1}{x} (\int \frac {1}{\sqrt {1-x^2}} dx) dx$ Now, $\int \frac {1}{\sqrt {1-x^2}} ...
4
votes
3answers
67 views

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

Integrate using integrating by parts : $$\int \frac{(1-x)^2 \cdot e^x}{(1+x^2)^2}dx$$ My attempt : $$I=\int \frac{(1-x)^2 \cdot e^x}{(1+x^2)^2}dx$$ $$I=\int \frac{e^x}{(1+x^2)^2}\cdot(1-x)^...
1
vote
3answers
67 views

Solve $ \int{x\sin^2(x)}\ dx $

I need to solve this integral: $$ \int{x\sin^2(x)}\ dx $$ I SOLVED it by writting: $$ \sin^2(x) = \frac{1-\cos(2x)}{2} $$ and used integration by parts for $x\cos(2x)$, and the result is: $$ \frac{1}...
4
votes
3answers
57 views

integrate $\int \frac{x^4-16}{x^3+4x^2+8x}dx$

$$\int \frac{x^4-16}{x^3+4x^2+8x}dx$$ So I first started with be dividing $p(x)$ with $q(x)$ and got: $$\int x-4+\frac{8x^2+32x-16}{x^3+4x^2+8x}dx=\frac{x^2}{2}-4x+\int \frac{8x^2+32x-16}{x^3+4x^2+...
2
votes
1answer
48 views

Why $C=1$ in indefinite integral $\int{\sin x\,\mathrm{d}x}$

I am reading Introduction to Calculus and Analysis by Richard Courant. In Section 3.15, g.The Dirichlet Integral, it said $\int{\sin x \,\mathrm{d}x}=1-\cos x$,why $C=1$ here?
-1
votes
2answers
93 views

2 Weird questions

Seems like I'm full of weird mathematical questions! Last time I made a question about imaginary numbers. This time I have 2 seemingly unrelated questions. But nevertheless it's always good (and fun)...
5
votes
1answer
77 views

Finding $\int \frac{\mathrm{d}x}{1 + \frac{2}{x} - x}$

I want to solve: $$\int\frac{1}{1+\frac{2}{x}-x} \mathrm{d}x $$ I don't know how to start, maybe I should use partial fraction?
0
votes
3answers
84 views

Ambiguous integral

What is the real integral of the function $$f(x) =\frac{1-x^2}{(1 + x^2)^2}$$? Is it $F_1(x) = \frac{x}{1 + x^2} + C$ or $F_2(x) = \arctan x + C$ ? The brochure I was reading gave the first result ...
4
votes
4answers
155 views

Find $\displaystyle \int{\dfrac{1}{\sqrt{1+x^4}}dx}$

Find $\displaystyle \int{\dfrac{1}{\sqrt{1+x^4}}dx}$ Let $x^2=\tan u$ $\implies 2x dx=\sec^2 u du$ $\implies dx=\dfrac{\sec^2 u}{2\sqrt{\tan u}}du$ $=\displaystyle \int{\dfrac{\sec^2 u}{2\sec u\...
1
vote
2answers
99 views

A Tangent Integral $\int{\frac {1+\tan^2 (x)}{1+\tan^5 (x)}}dx$

What is the answer the following integral? $$I=\int{\frac {1+\tan^2 (x)}{1+\tan^5 (x)}}dx$$ It is sufficient to factor the polynomial $x^5+1$.
1
vote
0answers
50 views

Solve $ \int{\sqrt{1 + (3x^2 + 2x - \frac{29}{2})^2}} dx $

I have to solve this indefine integral: $$ \int{\sqrt{1 + (3x^2 + 2x - \frac{29}{2})^2}} dx $$ I tried to make the square: $$ \int{\sqrt{9x^4 +12x^3-29*3x^2 -58x + \frac{29^2 +4}{4}}} dx $$ but ...
0
votes
1answer
56 views

Question about integration by parts

Could someone please explain how we calculated the first term of the second identity? Thanks a ton.
0
votes
3answers
87 views

Evaluate $\int \frac{\sqrt{64x^2-256}}{x}\,dx$

$$\int \frac{\sqrt{64x^2-256}}{x}\,dx$$ Image. I've tried this problem multiple times and cant seem to find where I made a mistake. If someone could please help explain where I went wrong I would ...
-1
votes
3answers
71 views

How to present this particular integration [closed]

$$ \int\frac{x^2+3x}{\sqrt{x^2+6x+10}}dx $$ How to present this integral?
4
votes
3answers
503 views

Indefinite Integral - How to do questions with square roots?

$$\int \frac{dx}{x^4 \sqrt{a^2 + x^2}}$$ In the above question, my first step would be to try and get out of the square root, so I would take $ t^2 = a^2 + x^2 $. But that gets me nowhere. How ...
4
votes
2answers
98 views

Finding the integral $\int \frac{2x^{12} + 5x^9}{(x^5 + x^3 + 1)^3}dx$ with substitution - how to think?

Find $$\int \frac{2x^{12} + 5x^9}{(x^5 + x^3 + 1)^3}dx$$ In the above question, I was literally stumped, and wasn't able to solve it for a long time. Turns out that you had to divide the ...
2
votes
2answers
30 views

Integrating functions with algebraic and trigonometric parts. $\int\frac{x}{\sec x + 1}dx$

$$\int\frac{x}{\sec x + 1}dx$$ How to perform this integration? I tried simplifying it to $$\frac{x \cos x}{1 + \cos x}$$ but after that integration by parts is not useful.
1
vote
1answer
35 views

Having trouble understanding the output of the integral

Exposure is given by $$E=\max(V,0)=\max(\mu+\sigma Z,0)$$ The EE defines the expected value over the positive future values and is therefore:$$\mathbb{E}[E]=\int_{-\mu/\sigma}^{\infty}(\mu+\sigma x)\...
2
votes
5answers
160 views

Integrate $\int \frac{x^2-2}{(x^2+2)^3}dx$

$$\int \frac{x^2-2}{(x^2+2)^3}dx$$ What I did : Method $(1) $ Re writing $x^2-2 = (x^2+2)-4 $ and partial fractions. Method $(2) $ Substituting $x^2 = 2\tan^2 \theta $ Is there any other easy ...
1
vote
1answer
87 views

how to solve $ \int \frac{ e^{4cy^3+2by^2 + (a-3c)y - b}} {\sqrt{1-y^2}} dy $?

here $a,b,c$ are constants it can be solved as indefinite integral or a definite integral with limits [-1,1] or [0,1] MATLAB is not helping here
1
vote
2answers
46 views

I have trouble understanding when I can simply take the anti-derivative of square root functions and when I have to solve by other means.

I don't know when it is and when it is not okay to simply take the anti-derivative of a square root function when I'm trying to find the integral. Could someone explain to me when and when it is not ...
1
vote
1answer
60 views

Can't get rid of integrals solving this differential equation.

Assuming $\mathbf{A}\equiv \vec A$ , $\dot q\equiv \frac{d}{dt}q$ ,and $\ddot q\equiv \frac{d^2}{dt^2}q$ , And Using the Right-hand Cartesian coordinate system with base vectors $\mathbf{\hat i\, \...
1
vote
2answers
58 views

How to perform the following integration $\int \frac{\cos 5x+5\cos 3x+10\cos x}{\cos 6x+ 6\cos 4x+ 15\cos 2x +10}dx$

$$\int \frac{\cos 5x+5\cos 3x+10\cos x}{\cos 6x+ 6\cos 4x+ 15\cos 2x +10}dx$$ How to simplify the expression given? I tried using formulas for $\cos 2x$ and $\cos 3x$. I also tried the using $\cos x +...
3
votes
3answers
777 views

Is this an elementary integral?

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
56 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
77 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
677 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
240 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
84 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
40 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
17 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
22 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
209 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?