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

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0
votes
3answers
43 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 ...
0
votes
0answers
17 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} ...
0
votes
1answer
24 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
218 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 ...
0
votes
2answers
74 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: ...
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
69 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 ...
5
votes
3answers
105 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
41 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 ...
0
votes
0answers
35 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 ...
1
vote
1answer
56 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 = ...
1
vote
4answers
126 views

Error in solving $\int \sqrt{1 + e^x} dx$ .

I want to solve this integral for $1 + e^x \ge 0$ $$\int \sqrt{1 + e^x} dx$$ I start by parts $$\int \sqrt{1 + e^x} dx = x\sqrt{1 + e^x} - \int x \frac{e^x}{2\sqrt{1 + e^x}} dx $$ Substitute ...
2
votes
0answers
85 views

Integrate $\cos^2(\pi x)\cos^2(\frac{n\pi}{x})$

I need to integrate $\int\cos^2(\pi x)\cos^2(\frac{n\pi}{x})dx$. There's no limit for $x$ but if it helps you can assume $\frac{n}{10} \le x \le n$. Walfram calculator can gives the following ...
1
vote
0answers
27 views

Indefinite integral involving exponential and arbitrary function

If I have for any piecewise continuous function $f$ $$ F(x) = \int f(x)\ dx $$ is it possible to calculate $$ \int f(ax+b)\ e^{cx}\ dx $$ If $f$ is $sin$ then a computer algebra system gives me $$ ...
1
vote
3answers
36 views

Approximation of an indefinite integral

Consider this integral $$\frac{1}{2d}\int_{-d}^{d}f(x-t) \, \mathrm{d}t$$ When $d$ goes to zero, $$\lim _{d\to 0} \frac{1}{2d}\int_{-d}^{d}f(x-t) \, \mathrm{d}t = f(x)$$ but what is the second ...
0
votes
2answers
67 views

Integration of $f(x)^n$, given $f(x)$ is continuous and $0 \le f(x) \le 1$

Given $f(x)$ is continuous and $0 \le f(x) \le 1$, and $\int f(x) dx = g(x)$, is there indefinite or definite integration formula for $$\int f(x)^n dx$$ or an approximation or expansion?
2
votes
2answers
69 views

Integrate $ \int\frac{1+x^2}{(1-x^2)(\sqrt{1+x^4})}dx $

$$ \int\frac{1+x^2}{(1-x^2)(\sqrt{1+x^4})}dx $$ I thought of substituting $ x-\frac{1}{x} $ as $t$ but it gets stuck midway. I am close but I think I need to sustitute something else here.
-2
votes
1answer
40 views

Integrate $ \int\frac{x-1}{x^2\sqrt{2x^2-2x+1}}dx $ [closed]

$$ \int\frac{x-1}{x^2\sqrt{2x^2-2x+1}}dx $$ I am unable to begin this question. Please give some hints or solve it.
3
votes
1answer
93 views

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

$$ \int \frac{x^2 + x}{(e^x + x +1)^2}dx $$ I cant think of any substitution to start this question.
1
vote
2answers
39 views

Proving integration formula

I want to prove the integration formula $$ \int \frac {\sqrt {a+bu}}{u} \ du = 2 \sqrt {a+bu}+a \int { \frac {du}{u \sqrt {a+bu}} }. $$ I tried trigonometric substitution (as $u= \frac {a \tan^2 ...
1
vote
5answers
110 views

Integrate $ \int \frac{1}{1 + x^3}dx $

$$ \int \frac{1}{1 + x^3}dx $$ Attempt: I added and subtracted $x^3$ in the numerator but after a little solving I can't get through.
3
votes
3answers
59 views

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

$$ \int \frac{1+x\cos x}{x(1-x^2e^{2\sin x})}dx $$ Attempt: I substituted $(1-xe^{2sin(x)})$ by $u$ and tried from there by differentiating it. But I get stuck midway.
-1
votes
1answer
62 views

Evaluating $\int\frac{x^4}{x^2+a^2}dx$ [closed]

Please how to integrate the following function? $$ \frac{x^4}{x^2+a^2} $$
1
vote
2answers
74 views

Deadly integral

How to solve this question $\int\limits_0^1\frac{x^{2}+x+1}{x^{4}+x^{3}+x^{2}+x+1}dx$ . Please help me in solving this short way my approach is in the answer Is it correct and can it be solved in ...
0
votes
1answer
35 views

Elementary Functions, Differentiation, Integration [duplicate]

Why is it that differentiation of a function that is a composition of elementary functions (such as $\sin \:2^x$ or $\ln(\mathrm{arcsec}\: x^3)$ or $x^{1/x}$) always produces a composition of ...
0
votes
3answers
60 views

Evaluate the algebraic indefinite integral $\int \frac{\sqrt{x+1}-\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}}dx$ [closed]

I have no idea how to solve it, can you show me how to start with this integral? $$\int \frac{\sqrt{x+1}-\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}}dx$$ Thank you.
3
votes
2answers
57 views

Indefinite integral of $\int \frac { dx}{x^{2m}+1}, m \in \mathbb R $

I spent a few hours trying to solve this indefinite integral: $$\int \frac { dx}{x^{2m}+1}, m \in \mathbb R $$ I tried to transform the fraction to partial fractions getting $\int ...
-2
votes
1answer
84 views

A Beautiful Integral: $\int_{0}^{\pi/2}\log(\sin x)\log(\cos x)\,dx$ [closed]

I have to find the value of $$\int_{0}^{\frac{\pi}{2}}\log(\cos(x))\log(\sin(x))dx$$ in terms of $\pi$ and $\log(2)$. Any hint?
0
votes
2answers
49 views

u substitution and du in integrals?

These were a couple examples in class before learning the $u$-substitution method for integrals. I'm not sure what is going on: $$\int1\, d(2x) = 2x+ C$$ $$\int \sin x \,d(\sin x) = \frac{(\sin ...
0
votes
3answers
60 views

How do I apply $u$-substitution to solve $\int x \ln\left(1+\sqrt{x}\right) \,dx$

$$\int x \ln\left(1+\sqrt{x}\right) \,dx$$ I think u-substitution would be the best method but I'm not entirely sure how I would go about using it.
2
votes
2answers
75 views

Integrate $\frac{\sin^3 \frac{x}{2}}{\cos \frac{x}{2} \sqrt{\cos x+\cos^2 x+\cos^3 x}}$

Evaluate $$\int \frac{\sin^3 \frac{x}{2}}{\cos \frac{x}{2} \sqrt{\cos x+\cos^2 x+\cos^3 x}}dx$$ I saw terms like $1+t+t^2$ in the denominator , so I thought of $t^3-1$ and then converting back into ...
0
votes
0answers
42 views

Integration of the function with good substitution

$$\int\frac{\ln(x)}{1+\ln(x)^2}\mathrm{d}x$$ I surely know the integral would be of $u/v$ type but I am not getting any good substitution to go for it. I think $\log(x)=e^t$ would be good as we get ...
1
vote
0answers
33 views

Evaluate $\int \frac{\sec x \:dx}{\sin (2x+\theta)+\sin \theta}$

Evaluate $$I=\int \frac{\sec x \:dx}{\sin (2x+\theta)+\sin \theta}$$ I used $$\sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)$$ we get $$I=\frac{1}{2} \times \int ...
1
vote
3answers
74 views

Integral of the function $\frac{\cos ^2 x}{1+\tan x}$

Evaluate $$\int \frac{\cos ^2 x}{1+\tan x}dx$$ I tried converting in double angle and making the derivative of the denominator in the numerator. But, it didn't work out. Some help please. Thanks.
4
votes
1answer
62 views

Integrate $\sin^2(nx)\csc^2(x)dx$

I need help to integrate $$\int{\sin^2(nx)\csc^2(x)dx}$$ Integral of each part is easy $$\int{\sin^2(nx)}dx=\frac{x}{2}+\frac{\sin(2nx)}{4n}+C$$ $$\int{\csc^2(x)}dx=-\cot(x)+C$$ I then tried to use ...
4
votes
1answer
114 views

Evaluating the integral $\int \frac{x^2+x}{(e^x+x+1)^2}dx$

Evaluate $$\int \frac{x^2+x}{(e^x+x+1)^2}dx$$ I tried converting in the form of Quotient rule(seeing the square in the denominator), neither am I able to make the denominators' derivative in the ...
0
votes
4answers
52 views

Integral of $\dfrac{1}{(x-a)(x-b)}$ when $a, b$ are both different and the same ($2$ questions)

So I was assigned a question in my Calc II class, and as I got into the problem, I got lost on what I was being asked to find. As I looked at the two integrals, I was able to split both up into ...
3
votes
7answers
758 views

Evaluating the rational integral $\int \frac{x^2+3}{x^6(x^2+1)}dx $

Evaluate $$\int \frac{x^2+3}{x^6(x^2+1)}dx .$$ I am unable to break into partial fractions so I don't think it is the way to go. Neither is $x=\tan \theta$ substitution. Please give some hints. ...
0
votes
0answers
24 views

Indefinite INTEGRAL fraction ( irrational ) [duplicate]

$$\int \frac{x-1} {1 + \sqrt{x^2+2x-3}}dx$$ The teacher advised me to calculate through Euler's substitution. Please help me to find the integral (similar topic was created earlier but now I need ...
1
vote
4answers
195 views

Question on Indefinite Integration: $\int\frac{2x^{12}+5x^9}{\left(x^5+x^3+1\right)^3}\,\mathrm{d}x$

Give me some hints to start with this problem: $${\displaystyle\int}\dfrac{2x^{12}+5x^9}{\left(x^5+x^3+1\right)^3}\,\mathrm{d}x$$
1
vote
5answers
64 views

A rational function integration

Evaluate $$\int \frac{3x^2+1}{(x^2-1)^3}dx$$ I tried breaking the numerator in terms of the denominator but it didn't help much. I also tried a few substitutions buy most of them were useless. Please ...
0
votes
1answer
22 views

Problem on Indefinite Integration (Calculus) [closed]

How do I solve $\displaystyle\int \dfrac{dx}{x^{2} (x^{4} + 1)^{\frac{3}{4}}}$.
0
votes
0answers
15 views

Questions about simplifying an equation and about finding orthogonal trajectories

First question: a question involves simplifying the following equation (or to be precise, making the equation an explicit function) $$8y^2=x^2(1-x^2).$$ When I tried to simplify it, I got $$y= \pm ...
0
votes
4answers
113 views

Have any meaning $\int_0^{x} x dt$? [closed]

If $x$ and $t$ are independent variables, does this expression have any meaning? $$\int_0^{x} x \hspace{2pt}\mathrm{d}t$$ I admit that I'm confused about this!
5
votes
2answers
34 views

Carrying out a substituting to evaluate $\int (x + 1) (x^2 + 2 x)^5dx$

The problem is: $$ \int { (x+1)({ x }^{ 2 } } +2x{ ) }^{ 5 }dx $$ The next step given by WolframAlpha is $$\int { (x+1)({ x }^{ 2 } } +2x{ ) }^{ 5 }dx\\ =\quad \frac { 1 }{ 2 } \int { { u }^{ 5 }du ...
1
vote
1answer
55 views

Integration of a polylogarithm: Is this function known?

I would like to integrate a polylogarithm of a given order $$\int dx \mbox{Li}_{n-1}(x)$$ suppose that the order is $n\le 0$ and $x\in(-\infty,0]$, so the function is bounded. I know that it can be ...
1
vote
4answers
79 views

$\int \frac{1}{\sqrt{x^2+1}} dx$

So I've seen some options on the internet that are fairly good, but I have this substitution: $x^2+1=t-x$, you square both sides and get $x = (t^2-1)/t$ and $x + 1 = (t^2-1)/2t + 1$. If we call that ...
0
votes
1answer
71 views

Hint for solving a indefinite integral [closed]

Can anyone provide a hint for solving this definite integral $$\int \dfrac{\sqrt{1-a^2 +x^2}}{x^2(a^2-x^2)}dx$$ Here $a$ is a real constant. I'll appreciate any help. Thanks.