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

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5
votes
1answer
56 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
56 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 } ...
1
vote
1answer
22 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
68 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 ...
5
votes
2answers
87 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
99 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
102 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
52 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
59 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
50 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
0answers
150 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
108 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 ...
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
96 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
65 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
62 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
90 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
29 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
102 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
45 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
62 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
150 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
66 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 ...
1
vote
3answers
52 views

I need help solving this indefinite integrals problem?

I am doing indefinite integrals homework, and this problem popped up. I hate to post on here without any personal insight on the problem, but I really have no idea on how to approach this.I do not ...
2
votes
1answer
79 views

Evaluate $\int\frac {\csc^2{x}-2005}{\cos^{2005}{x}} dx $

Evaluate the indefinite integral $$\int\frac {\csc^2{x}-2005}{\cos^{2005}{x}} dx$$ I tried multiplying and dividing by $\sec^2 {x} $ and then setting $\tan{x}=y$ but no good. Then I set $\cos ...
1
vote
1answer
89 views

Is this formula for the harmonic numbers true?

Is this formula for the harmonic numbers true? $$H_n = \lim_{s\to 0} \, \int \frac{(s+1)^{(-n-1)}+s-1}{s} \, ds$$ Mathematica: ...
3
votes
4answers
133 views

How would I go about evaluating $\int \frac{x}{(9-8x^2)^3}dx$?

So I have homework on webAssign (a site used by my college), and I am not understanding the logic as to why I am taking the steps into solving the integral it is telling me to take. So I'll list the ...
0
votes
2answers
46 views

What is happening to the '2' in this integral?

It is the indefinite integral: $\int \frac{1}{2x-6}$ I am trying to understand it and looking the last step goes from $\frac12 \log(2(x-3))$ to $\frac12 \log(x-3)$ Can someone explain to me why the ...
0
votes
3answers
97 views

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

How do I evaluate the following indefinite integral? $$\int x^2 \sqrt{x^2-1} dx$$ Through integration of parts, I have obtained $$ \frac{x}{3}(x^2-1)^{3/2} - \frac{1}{3} \int (x^2-1)^{3/2} dx $$ ...
0
votes
0answers
29 views

Integrating the logarithm of a function including a square root of a second degree polynomial

I have been trying for some time to calculate the following integral: $$\int \ln\left(k+\sqrt{ax^2+bx+c}\right)\ dx$$ where k, a, b and c are real numbers. I have tried several strategies, but without ...
-1
votes
0answers
25 views

changing the variable in integration [duplicate]

What is the way to change the variable with respect to we are doing integration For example in this integral [(cos2x)^1/2]dx/sinx How to change dx to d(cosx)
2
votes
3answers
185 views

Indefinite integral of trignometric function

What is the trick to integrate the following $$\int \frac{1-\cos x}{(1+\cos x)\cos x}\ dx$$
1
vote
4answers
497 views

How useful/useless is the indefinite integral

After having met yet another person confused by indefinite integrals today, I've finally decided to ask the community. Do you think it makes sense to teach indefinite integrals? My opinion is that ...
0
votes
0answers
57 views

Evaluate $\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx$ [duplicate]

As the title shown, how to evaluate the indefinite integral $$\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx\ ?$$ Thanks.
0
votes
0answers
62 views

Is the following integration of possible?

How to solve the following problem? $\int x^4/(1-x^4)^{3⁄2}dx$ I have tried the substitution, x=sinz, but failed.
2
votes
4answers
105 views

Antiderivative of $\frac{1}{1+\sin {x} +\cos {x}}$

How do we arrive at the following integral $$\displaystyle\int\dfrac{dx}{1+\sin {x}+\cos {x}}=\log {\left(\sin {\frac{x}{2}}+\cos {\frac{x}{2}}\right)}-\log {\left(\cos {\frac{x}{2}}\right)}+C\ ?$$
4
votes
2answers
142 views

Evaluation of $\int\frac{\sqrt{\cos 2x}}{\sin x}dx$

Evaluation of $$\displaystyle \int\frac{\sqrt{\cos 2x}}{\sin x}dx$$ $\bf{My\; Try::}$ Let $\displaystyle I = \int\frac{\sqrt{\cos 2x}}{\sin x}dx = \int\frac{\cos 2x}{\sin^2 x\sqrt{\cos 2x}}\sin xdx ...