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

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2
votes
3answers
101 views

What is $\operatorname{Ei}(x)$?

I was trying to solve $$\int\frac{e^x - e^{-x}}{x}\,dx$$ But I have no idea how to do it and the calculator said to use a common integral that I don't know what it means.
4
votes
2answers
60 views

Integral of the Square of the Elliptic Integral

Someone must know a good technique for $$ \int E^{2}(x)dx $$ Where $E$ is the complete elliptic integral of the second kind: $$ ...
2
votes
0answers
41 views

Simple Integral Involving the Square of the Elliptic Integral

I have, $$ \int uE^{2}\left(u\right)du $$ where $E$ is the complete elliptic integral of the second kind: $$ E\left(k\right)=\int_{0}^{\frac{\pi}{2}}d\theta\sqrt{1-k^{2}\sin^{2}\left(\theta\right)} ...
2
votes
1answer
54 views

Indefinite integration of $1/\sqrt{3-5x-2x^2}$

Cannot make it out. $$\int \frac{dx}{(3-5x-2x^2)^{1/2}} $$ Is the problem correct, or does it have errors? I have a doubt.
3
votes
1answer
65 views

How do I integrate $\int_{0}^{\frac{\pi^2}{4}}7\sin(\sqrt{x})dx$?

So, quick backstory. My semester just started and we are starting off by learning integration by parts. Which hasn't caused me much trouble except for this problem. ...
2
votes
1answer
44 views

solve this ordinary differential equation?

i have the differential equation $y'=\frac{y-x}{y-x+1}$, how i solve this? try: i tryed to substitute $u=y-x$, then $u=y-x\iff y=u+x\Rightarrow y'=u'+1$ then $y'=\frac{y-x}{y-x+1}$ become ...
3
votes
1answer
98 views

When may we ignore the limits of integration?

When we try to evaluate an integral such as, say $$\int_a^b{f(x)dx}$$ there is often the case that we can analytically find $$\int{f(x)dx}$$ a little faster (imagine leaving away the evaluation ...
2
votes
0answers
31 views

Tractable indefinite integral of the exponentiation of some function

Consider the function $z(s)\in\mathbb{C}$ defined as $z(s)=\int_0^s \exp\left[i(q u+\lambda(u))\right]du$ for some $q\in\mathbb{Q}-\mathbb{Z}$ and $\lambda(s)$ a $2\pi$-periodic real differentiable ...
10
votes
3answers
177 views

How do I integrate $\frac{\sqrt{1-k^2\sin^2 x}}{\sin x}$

How do I evaluate this indefinite integral, for $|k| < 1$: $$ \int\frac{\sqrt{1-k^2\sin^2 x}}{\sin x}\mathrm{d}x $$ I tried the change of variable $t=\sin x$, and obtained two integrals, but I ...
0
votes
1answer
39 views

Partial fraction decomposition and polynomials?

This answer gives a really great explanation of why partial fraction decomposition works. However, the explanation implies that rational functions can be decomposed into a sum of fractions plus a ...
0
votes
2answers
48 views

Integral of $\arcsin$ of a rational function, using integration by parts

I'm a class 12 student and this a question from my textbook: $$I=\int{\arcsin{2x\over 1+x^2}}\mathrm{d}x$$ I did it using integration by parts like this: $$I=\arcsin{\left(2x\over ...
2
votes
2answers
67 views

Finding substitution in the integral $\int{\frac{2+3x}{3-2x}}dx$

In a problem sheet I found the integral $$\int{\frac{2+3x}{3-2x}}dx.$$ In the solution the substitution $z=3-2x$ is given which yields $x=\frac{3-z}{2}$ and $dx=-\frac{1}{2}dz$. We have ...
5
votes
1answer
74 views

How to evaluate $\int \frac{\mathrm{dx}}{x^4[x(x^5-1)]^{1/3}}$

How to evaluate: $$\int \frac{\mathrm{dx}}{x^4[x(x^5-1)]^{1/3}}$$ I have done a substantial work on it: Let $x^5z^3=x^5-1$. So $$x^5(z^3-1)=1\implies ...
3
votes
4answers
71 views

Shorter way to integrate $\int \frac{x^9}{(x^2+4)^6} \, \mathrm{d}x$

$$ I=\int \frac{x^9}{(x^2+4)^6}\mathrm{d}x $$ Yeah I know, I can substitute: $$t=x^2+4\text{ or }2\tan\theta$$ So that: $$I=\frac12\int\frac{(t-4)^4}{t^6}\mathrm{d}t\text{ or } ...
8
votes
1answer
145 views

How to find $\int \frac{x^4-4}{x^2\sqrt{4+x^2+x^4}} \,\mathrm dx$

Integrate $$\int \frac{(x^4-4)}{(x^2\sqrt{4+x^2+x^4})}\mathrm dx$$ My try: $$\int \frac{(x^2-4/x^2)}{(\sqrt{4+x^2+x^4})}\mathrm dx\\ =\int \frac{ (x^2-4/x^2)}{(\sqrt{(x^2+1/2)^2+15/4})}\mathrm ...
1
vote
1answer
26 views

Indefinite integral: $\int \frac{\sqrt{x^2-6x+18}}{x-3}dx$

I have the following indefinite at hand and I'm sure substitution is the only way I should go about solving this, but each time I think I get close, I end up at the same place, which doesn't seem to ...
3
votes
1answer
70 views

How to find $\int \frac{\cos5x + 5\cos3x +10\cos x }{\cos6x+ 6\cos4x + 15\cos2x +10}\mathrm dx$

I have a integral which seems difficult to me. Any help would be appreciated. Find $$\int \frac{\cos5x + 5\cos3x +10\cos x }{\cos6x+ 6\cos4x + 15\cos2x +10}\mathrm dx$$ Also I wound like to know ...
4
votes
2answers
88 views

Evaluate $\int\frac{8x+20}{5x^2+25x+20}dx$

I tried to solve it and got $\frac{4}{5} \ln(4+5 x+x^2)+C$ as an answer, but my online homework program says it's incorrect. What did I do wrong? I pulled out $\frac{4}{5}$ as a constant and saw ...
7
votes
4answers
112 views

How to find $\int \frac{x\ln(x+\sqrt{1+x^2})}{\sqrt{1+x^2}}\mathrm dx$

$$I=\int x.\frac{\ln(x+\sqrt{1+x^2})}{\sqrt{1+x^2}}\mathrm dx$$ Try 1: Put $z= \ln(x+\sqrt{1+x^2})$, $\mathrm dz=1/\sqrt{1+x^2}\mathrm dx$ $$I=\int \underbrace{x}_{\mathbb u}\underbrace{z}_{\mathbb ...
1
vote
2answers
103 views

How to find $\int {t^n \, e^{t}}\mathrm dt$?

Consider:$$\int {t^n e^{t}}\ \mathrm dt$$ is there any closed formula for this? W|A gave me this but I don't know what is Gamma function: $$\int {t^n e^t\ \mathrm dt} = (-t)^{-n}\ t^n\ \Gamma(n+1, ...
2
votes
1answer
55 views

How to solve $\int \frac{\tan^{-1}x}{(1+x)^2}dx$?

I know how to solve the following integral $$\int \frac{\tan^{-1}x}{(1+x^2)}dx$$ . We have to substitute $\tan^{-1}x$ as $t$ and we will be done. After this one, I tried to find out $$\int ...
0
votes
1answer
61 views

Integrals related to the function $F(x) = \int_1^x (e^t/t )\, dt$

I'm having some trouble with part of a problem from Apostol Volume 1(Section 6.26, Number 6). For completeness I'll include the whole question: A function $F$ is defined by the following indefinite ...
4
votes
4answers
100 views

Evaluate $\int{\sin^3(x)\cos^2(x)}dx$

I'm trying to solve $\int{\sin^3(x)\cos^2(x)}dx$. I got $-\frac{1}{2}\cos(x)+C$, but the memo says $\frac{1}{5}\cos^5(x)-\frac{1}{3}\cos^3(x)+C$ This is my working: Your help is appreciated!
1
vote
0answers
52 views

Integration of log(sin(x)) [duplicate]

Can anyone please help me the following indefinite integral: $$\int \log(\sin(x))dx$$ Thanks
3
votes
1answer
92 views

How can I do this? $\int\frac{dx}{x^4+1}$ [duplicate]

I tried to integrate this: $\displaystyle\int \dfrac{dx}{x^4+1}$ I tried to do it with the partial fractions method (after factoring the denominator), but the process is really large, and I got a lot ...
6
votes
1answer
175 views

Integration of combination of Bessel Function and Exponential Function

I have read "Watson:Treatise Theory of Bessel Function", "Table of Integration, Series and Product", "Handbook of Mathematical Functions, Formulas, Graphs and Mathematical Tables" and other online ...
1
vote
1answer
110 views

How can I express $\int \frac{1}{f'(x)}$ in terms of $f(x)$

More specifically, I would like to know if there is a way I can express $$\int \frac{x g'(x)}{f'(x)} dx $$ In terms of $f(x)$ and $g(x)$. Both $f(x)$ and $g(x)$ are non-negative and known to be ...
3
votes
0answers
50 views

Integration indefinite integral of multiple functions

I need help integrating $$\frac{x}{1-\exp(-x^2/a^2)}\exp((x-u)^2/2s^2)$$ wrt $x$, where $a$ and $u$ are constants
0
votes
2answers
43 views

Calculus long division $\int\frac{y^4+3y^2-1}{y^3+3y}\ dy$

I have a problem like this in my homework and want to see how to go by doing this problem. I understand the long division, but cannot get the partial fraction part. $$\int\frac{y^4+3y^2-1}{y^3+3y}\ ...
1
vote
1answer
97 views

Can you find integral of this function.

Question: Consider $$F(x) = \frac{1}{\sin(x-a) \ \sin(x-b) \ \sin(x-c)}$$ Then, how to compute $\int F(x) \, \mathrm{d}x$? Edit: I have tried what I know about integral solving methods. I ...
4
votes
4answers
66 views

Integration of some floor functions

Can anyone please answer the following questions ? 1) $\int$ $ \left \lfloor{x}\right \rfloor $ $dx$ 2) $\int$ $ \left \lfloor{\sin(x)}\right \rfloor $ $dx$ 3) $\int_0^2$ $\left ...
1
vote
2answers
70 views

An intergral with variable upper limit

Let $$\psi \left(x \right)=\int_{0}^{x}\frac{\ln(1-t)}{t}dt,x\in (0,1).$$ Show $$\forall x\in (0,1), \psi\left(x \right)=?$$ I return the old variable $t$ by the substitution $s=ln(1-t)$,and then ...
-2
votes
1answer
63 views

Some confusing and tough (for me) integrations [closed]

Can anyone please help me with these integrations : $\int_0^3$ $| x+1 |$ $dx$ $\int$ $(|x-2|+|x-1|+|x|+|x+1|+|x+2|)$ $dx$ $\int$ $|x|dx$ $\int$ $(e^{|x|}$ + $\ln x)$ $dx$ $\int_0^\pi$ ...
0
votes
1answer
92 views

Calculate the integral of $\sqrt{36\sin^2(2t)+6\cos^2(t)}$

During an arc length calculation I reached the following integral and I am having hard time calculating it: $$\int\sqrt{36\sin^2(2t)+6\cos^2(t)}\,dt=\sqrt{6}\int\cos t \sqrt{24\sin^2(t)+1}\;dt$$ ...
2
votes
0answers
30 views

Solving $n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt$

I have to solve $$ n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt $$ where $\psi(t)=(2\pi)^{-\frac{1}{2}}e^{-\frac{1}{2}t^2}$ is the density ...
2
votes
3answers
102 views

About integrating $\sin^2 x$ by parts

This is about that old chestnut, $\newcommand{\d}{\mathrm{d}} \int \sin^2 x\,\d x$. OK, I know that ordinarily you're supposed to use the identity $\sin^2 x = (1 - \cos 2x)/2$ and integrating that ...
3
votes
3answers
445 views

How to find the antiderivative of $\frac{1}{x^2(1+x^2)}$?

How to find the antiderivative of $\dfrac{1}{x^2(1+x^2)}$? I recognized that this can be done with trigonometric substitution and I let $x = \tan(x)$ and ended up with $\dfrac{1}{\tan(x^2)}$; then I ...
0
votes
2answers
103 views

Evaluate $\int \frac{du}{(u^2+2)^2}$ [closed]

Someone can help me with some idea to solve the integrate $$\int \frac{du}{(u^2+2)^2}$$ I tried to solve it using trigonometric substitution, but it failed.
2
votes
2answers
130 views

How to evaluate the following indefinite integral? $\int e^{e^x}\mathrm dx$

I stumbled across this question: what's the value of the following integral? $$\int e^{e^x}\mathrm dx.$$ Furthermore I was required to demonstrate. On wolfram I got the result $\operatorname{Ei}(e^x)$ ...
1
vote
1answer
46 views

$\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx<\infty $ for some large $n$?

Fix $y\in \mathbb R.$ Define, $$I(y)=\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx.$$ My Question is: Can we show that $I(y)<\infty$ for some large $n\in \mathbb N$ ? If yes, what is a value ...
1
vote
0answers
42 views

$\int_{0}^{y} \exp\left(\, -\alpha \sqrt{x(1-x)}\,\right)\, {\rm d}x = \int_{0}^{y} \exp\left(\, -\alpha \sqrt{x}\,\right)\, {\rm d}x$?

Are the following integrals equal for large $\alpha$: $$ I_1 =\int_{0}^{y} \exp\left(\, -\alpha \sqrt{x(1-x)}\,\right)\, {\rm d}x $$ $$ I_2 =\int_{0}^{y} \exp\left(\, -\alpha \sqrt{x}\,\right)\, {\rm ...
0
votes
2answers
46 views

Justification for U-substitution method

I am currently learning how to find antiderivatives using the "u-substitution" or "integration by substitution" method. A key component of this is setting some expression in the indefinite integral as ...
1
vote
1answer
77 views

Tough integral with many radicals

I am completed baffled with this integral $$\int\left[\dfrac{1}{x^{1/3}+x^{1/4}}+\dfrac{\ln(1+x^{1/6})}{x^{1/3}+x^{1/2}}\right]\mathrm dx$$ Any tips?
1
vote
0answers
67 views

Verification of Fourier transformation of Io-sinh function

I try to match, but it could not match $I_o-\sinh$ Practical Fourier Transform pair developed by Ben Logan, transform pair also published in The Practical Application of the Fourier Integral ...
3
votes
4answers
118 views

Evaluate $\int{\frac{xe^x}{(1+x)^2} dx}$

How would I evaluate this integral? $$\int{\frac{xe^x}{(1+x)^2} dx}$$ I know I need to use parts but I ended up getting a very complicated expression to integrate the second time.
3
votes
3answers
63 views

How do I solve $\int\frac{7}{\sqrt{x}(x+4)}~\mathrm{d}x$?

I am trying to solve $\int\frac{7}{\sqrt{x}(x+4)}~\mathrm{d}x$. So far I have $$7\int\frac{1}{\sqrt{x}(x+4)}~\mathrm{d}x$$ $$u=\sqrt{x}$$$$\mathrm{d}u=\frac{1}{2\sqrt{x}}$$ and this is where I'm not ...
4
votes
7answers
152 views

How do I go from this $\frac{x^2-3}{x^2+1}$ to $1-\frac{4}{x^2+1}$?

So I am doing $\int\frac{x^2-3}{x^2+1}dx$ and on wolfram alpha it says the first step is to do "long division" and goes from $\frac{x^2-3}{x^2+1}$ to $1-\frac{4}{x^2+1}$. That made the integral much ...
1
vote
3answers
45 views

Integrals with u substitution

Can someone please explain how the integration step highlighted in the red rectangle was worked out?
0
votes
0answers
67 views

Indefinite integral $\int\frac{e^x}{x(1+\log(x))}dx$

How to integrate this integral $$\int\frac{e^x}{x(1+\log(x))}dx$$ My attempt: I try some subtitutions, $e^x=u$$\hspace{0.2cm}$ and $\hspace{0.2cm}$$1+\log(x)=u$ but these are not helpful.Please help ...
2
votes
3answers
72 views

Trigonometric integral evaluation: $\int 4 \sin^4 x \cos^3 x \,dx$ [duplicate]

Evaluate the following integral $$\int 4 \sin^4 x \cos^3 x \,dx$$ I can do simple integration problems, but problems like this seem to stump me, I created this problem so I could solve and compare it ...