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

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1
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1answer
26 views

Are the real product rule and quotient rule for integration already known?

In "A Quotient Rule Integration by Parts Formula", the authoress integrates the product rule of differentiation and gets the known formula for integration by parts: \begin{equation}\int ...
0
votes
3answers
75 views

Integrate $\int{\frac{1}{(x+1)(x+2)^2(x+3)^3}dx}$

How to integrate this $$\int{\frac{1}{(x+1)(x+2)^2(x+3)^3}dx}$$ I tried to use that $$\int{\frac{1}{(x+1)(x+2)^2(x+3)^3}dx} = P_{1}(x)/Q_{1}(x) + \int{P_{2}(x)/Q_{2}(x)dx}$$ where ...
0
votes
0answers
63 views

Solving an indefinite integral problem [on hold]

The given problem is $$ \int \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx$$ please help me solving this indefinite integral problem...thank you very much, actually I have solved an definite integral ...
2
votes
1answer
53 views

Possibility of a closed form for $I_n = \int \frac{x^n e^{\tan^{-1}{x}}}{\sqrt{x^2+1}}\text{d}x$ where $n$ is a given integer.

The case of $n=2$ possesses an elementary closed form, which Mathematica 9 failed to find. This gives me an inkling of hope for the specific cases of the general form, as I have determined the ...
1
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1answer
67 views

Integrate $\int{ \left( \frac{1-x}{1+x} \right)^\frac{3}{2}dx}$

Integrate $$\int{ \left(\frac{1-x}{1+x} \right)^\frac{3}{2}dx}$$ I guess that there is sub $x = \cos t$ so integral gets to $$\int{ \left(\tan \frac{t}{2} \right)^3 d\cos t}$$ then I used that $\sin t ...
2
votes
4answers
72 views

Clever way of calculating the integral $ \int \frac{dt}{t^2\sqrt{t-2} } $

$$ \int \frac{\text{d}t}{t^2\sqrt{t-2} } $$ I know it can be calculated using somewhat complicated substitutions, but is there possibly some clever way of solving that type of integral? I don't ...
0
votes
2answers
55 views

Indefinite trignometric integral

I tried $u$-substitution and $uv$-substitution, can't seem to figure this out... any help would be appreciated! Question: $$\int\frac{x}{\cos(x)}\,dx$$ Thanks!!!
3
votes
2answers
83 views

How to solve integrals where you can't factor a polynomial?

Hi there guys I don't know if the title of the question should be the one for this but the thing is that I'm trying to solve this integral $\int \frac {\frac 12-u^2}{2u^4-2u^2+1}$$du$ and I have this ...
0
votes
1answer
16 views

Numerical integration of $E_1(x)$

I want to solve the following integral for $\gamma_0$: $$\int_{\gamma_0}^\infty \frac{1}{t}e^{-at} dt = c$$ for the specific values $a = 0.01$ and $c = 12.1$. As I understand, this is a variant of ...
-1
votes
3answers
63 views

Problem with Indefinite Integral $\int \frac {\cos^5x}{ 16(\cos^4x+\sin^4x)}dx$

Hello guys I'm totally lost in this indefinite integral, i'm just looking for advices/tips $$\int \frac {\cos^5x}{ 16(\cos^4x+\sin^4x)}dx$$ Should I begin with universal substitution? or there is ...
1
vote
2answers
42 views

How can I solve a first order ODE with $\pm$ signs by the Integrating Factor method?

I have the following first order ODE to be solved via the integrating factor method: $$\frac{\mathrm{d}z}{\mathrm{d}y}\pm z=-\frac12y\tag{1}$$ This is in the general form: ...
1
vote
1answer
37 views

Evaluate $\int{(2008 x^{2009}+2009x^{2007})}\cos(2008x) dx$

$$\int{(2008 x^{2009}+2009x^{2007})}\cos(2008x) dx$$ I tried to integrate the second term by parts and also the first.None of the terms seems to cancel out.I don't know why by parts is not ...
0
votes
0answers
8 views

Gaussian weighted intergal of Product of Gaussians

I'm trying to find a solution to the following function, My understanding is that the resultant function should still be a Gaussian, however I would like to define it as a linear function the ...
0
votes
1answer
34 views

Evaluate $\int ( \frac{1}{\sin x} - \sin x - \sin x \log(\sin x) ) dx$

This one looks easy but I still could'nt figure it out. $$\int ( \frac{1}{\sin x} - \sin x - \sin x \log(\sin x) ) dx$$ I tried substituting $\log(\sin x)=z$ but that's not working.Any suggestions?
2
votes
1answer
30 views

Finding integrals of the form $\int\frac{1}{x^2(x^{2009}+1)^{ \frac{2008}{2009}}}dx$

I faced two similar integrals today. They are $$\int\frac{1}{x^2(x^{2009}+1)^{ \frac{2008}{2009}}}dx$$ and $$\int\frac{1}{x^2(x^{2009}+1)^{ \frac{1}{2009}}}dx$$ No trigonometric substitution is ...
0
votes
1answer
52 views

How to find integral of the form $e^xf(x)$?

I always face trouble with these type of integrals. I need to find $$\int{e^x \frac{x(\cos x -\sin x)-\sin x}{x^2}}dx$$ My problem would be solved if can express $f(x)$ like $g(x)+g'(x)$ but ...
1
vote
1answer
35 views

Finding $\int e^{-\sin^2x}{(\cos x -3 x \sin(x)+2 x \sin^3(x))}dx$

I need to find $$\int e^{-\sin^2x}{(\cos x -3 x \sin(x)+2 x \sin^3(x))}dx$$ . I know that $$\int e^{g(x)}{(f(x)g'(x)+f'(x))}dx = e^{g(x)}f(x)$$. But I cannot find cannot $f(x)$ in the above ...
2
votes
2answers
72 views

integrate $\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6}dx$

$$\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6}dx$$ $$\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6}dx=\int \frac{3\left(\frac{16}{9}-x^2\right)^{\frac{3}{2}}}{x^6}dx$$ $x=\frac{4}{3}\sin\theta$ ...
1
vote
2answers
50 views

Problem with Indefinite Integral $\int\frac {\cos^4x}{\sin^3x} dx$

I'm stuck with this integral $\int\frac {\cos^4x}{\sin^3x} dx$ which I rewrote as $\int \csc^3x \cos^4xdx$ then after using the half angle formula twice for $\cos^4x$ I got this $\frac 14\int ...
2
votes
4answers
62 views

compute the value of an indefinite integral

Help me please with this indefinite trigonometric integral. How can I solve this kind of integrals? $$\int\limits \frac{1}{\left(\cos^4(x) \cdot \sin^2(x)\right)}dx$$
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votes
0answers
69 views

the integral of exponential function [closed]

all! How to calculate the integral of the following function $$\displaystyle\int_{- \infty}^{\infty} \displaystyle\frac{0.8e^{- ax^2 -ibx}}{1 -0.2 e^{icx}}dx $$ for $a,b,c \in \mathbb{R}$ and $i$ the ...
1
vote
1answer
48 views

Weird indefinite integral homework questions

I'm solving a couple of integration problems using the method of changing variables, and would like assistance with two particular problems that I can't seem to solve. I completed rest of the problems ...
2
votes
1answer
64 views

If $y'+y=|x|$ and $y(-1)=0$, what is $y(1)$?

If $y'+y=|x|$ and $y(-1)=0$, what is $y(1)$? I calculated the integrating factor to be $e^x$. Then $e^x y'+ e^x y=e^x |x|$ hence $\frac {d(e^x y)}{dx}=e^x |x|$ hence $d(e^x y)=e^x|x|dx $ ...
1
vote
2answers
31 views

Find volume of these solids using integration

a) The $(x>0, y< -1)$ region of the curve $y= -\frac{1}{x}$ rotated about the $y$-axis. The instructions say that one should use the formula: $V = \int 2πxf(x) dx$ I used another method and ...
0
votes
0answers
54 views

Integration of $\frac{x^2}{2\left(e^x+1\right)}$

Let: $$f(x) = \int \frac{x^2}{2\left(e^x+1\right)}dx $$ Is there a way to find $f(x)$? I've tried through integration by parts, but that didn't work out. If substitution is the answer, I can't see ...
1
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2answers
26 views

Trouble understanding solving integrals like linear equations

I know that some integrals on solving by parts end up with the same integral on the right side, and then the integral is assumed to be $I$ or some variable, and then linearly solved. For example, $$I ...
2
votes
3answers
27 views

Substitution and Partial Fractions (Integration)

$$\int\frac{dx}{x-\sqrt[4]{x}}$$ given the substitution $x=u^{4}, dx=4u^{3}du$ $$=\int\frac{4u^{3}du}{u^{4}-u}=\int\frac{4u^{3}du}{u(u^{3}-1)}=\int\frac{4u^{2}du}{(u^{3}-1)}$$ At this point I ...
7
votes
2answers
59 views

Is there no formula for $\cos(x^2)$?

I was wondering if there was a "formula" or an "identity" for $\cos(x^2)$, as there is for $\cos(2x)$. My question is closely related to this one, which was only asking for $\cos(ab)$. For instance ...
1
vote
3answers
90 views

How to solve $\int \frac{1}{1-y^2}$ with respect to $y$?

I was solving an A Level paper when I came across this question. I tried substitution, but I'm not getting the answer with that. Would appreciate it if someone would help me.
1
vote
1answer
36 views

How to use Substitution in an Abstract sense?

Based on my previous question: (somewhat related to it) $$\int f''(x^2)~dx$$ How would you go about and find the integral in an abstract sense as you can do the following with derivatives using ...
0
votes
0answers
20 views

Difference between definity integral with constants and with variables

What is general difference between: $$\int_a^b f(x) \;\mathrm{dx}$$ $$\int_a^{x^2} f(x) \;\mathrm{dx}$$
0
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2answers
40 views

Integration by Parts? - Variable Manipulation

$$\int x^3f''(x^2)\,\mathrm{d}x$$ Solve using Integration by Parts. \begin{align} u&=x^3\qquad\mathrm{d}v=f''(x^2) \\ \mathrm{d}u&=3x^2\qquad v=f'(x^2) \\ &=x^3f'(x)-\int f'(x^2)3x^2 ...
1
vote
2answers
22 views

Integral of $-4\sin(2t - (pi/2)) $ weird behavior on wolfram alpha

I'm confused by what Wolfram Alpha is doing with my function: $$-4\sin{(2t - (\pi/2))}$$ on why the it gets replaced by $$4\cos{(2t)}$$. Is it equal? Link: See behavior here
1
vote
4answers
46 views

Trigonometric Substitution (integration)

$$\int\frac{dx}{(x^{2}-36)^{3/2}}$$ My attempt: the factor in the denominator implies $$x^{2}-36=x^{2}-6^{2}$$ substituting $x=6\sec\theta$, noting that $dx=6\tan\theta \sec\theta$ ...
0
votes
2answers
40 views

Elementary integral for square roots of trig functions?

What's an easy way to calculate something like $\int \sqrt{1+\cos x} \text{ d}x$?
0
votes
0answers
16 views

Fourier transform of windowed complex exponential

I have a function on the form $$f(x) = g'(x)*e^{i\pi g(x)}.$$ Where $g'(x)$ is a window function with support in the range $-R \ldots R$. I want to find the fourier transform $\mathcal F(\omega)$ ...
1
vote
2answers
42 views

How to find the area for the curve $y=\sin^3(2x)\cos^3(2x)$?

I could calculate the integration of this by substituting $u=\sin(2x)$ and could find one of the limits of integration which was $0$. However, I couldn't find second limit. The mark scheme says the ...
2
votes
3answers
93 views

$ \int\frac{\sin(nx) \sin x}{1-\cos x} \,dx$ by elementary methods

What is an elementary way to show that for positive integer $n$ $$ \int\frac{\sin(nx) \sin x}{1-\cos x} \,dx= x + \frac{\sin (nx)}{n} + 2 \sum_{k=1}^{n-1}\frac{\sin(kx)}{k} $$ This cropped up when ...
0
votes
4answers
34 views

u-substitution, indefinite integrals

I've looked on the web for an answer to this question, and could not find an example. Could you push me towards a proper u substitution for the following integral? Please don't solve the problem just ...
0
votes
3answers
35 views

Setting up this integral?

$$\int\frac{2x+1}{9+x^2}dx$$ I tried to factor out the 9 to get $9(1+\frac{x^2}{9})=9(1+(\frac{x}{3})^2)$ to set up a u-sub to get arctan(x)..... But, it doesn't fit. Is this integration by parts? ...
0
votes
1answer
46 views

Evaluating the integral $ \int \bigl(\bigl(1-\frac{1}{2}z^2\bigr)^{-2}-1\bigr)^{-1/2} dz$ involved in the Young–Laplace equation

Through working on the Young-Laplace equation I cam across the following integral and Maple is acting strange: $$ \int{\frac{1}{\sqrt{\frac{1}{\left(1-\frac{z^2}{2}\right)^2}-1}} \, dz} .$$ If ...
0
votes
2answers
48 views

How do I perform u-substitution on this problem?

I am having trouble with this problem: $$\int {\frac{3x + 5}{5x^2 - 4x - 1}} dx$$ I can't seem to find a u where the du exists in the numerator so that it will cancel. If I choose: $$u = 5x^2 - 4x - ...
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votes
1answer
68 views

How would you integrate $1+\ln x$? [closed]

Could someone please show me how to integrate $1+\ln x$?
4
votes
4answers
67 views

Why can't you apply the natural logarithm rule to integrate $\int \frac{1}{\sqrt{x}}dx$?

I understand that $\frac{1}{\sqrt{x}} = x^{-\frac{1}{2}}$, or $\frac{1}{x^2} = x^{-2}$, but why wouldn't you be able anyhow able to apply the rule for which $\int \frac{1}{x}dx = \ln{|x|} + C$, and ...
0
votes
3answers
53 views

Possible evaluation of an indefinite integral $\int{1\over (x+a)^2\cdot(x+b)^2}$

Taking the integral: $$\int{1\over (x+a)^2\cdot(x+b)^2}$$ I tried to rewrite it in such a way: $$\int{1\over a-b}\cdot{1\over 2x+a+b}\cdot{(x+a)^2-(x+b)^2\over (x+a)^2\cdot(x+b)^2}$$ I have no idea ...
2
votes
3answers
55 views

Choosing a substitution to evaluate $\int \frac{x+3}{\sqrt{x+2}}dx$

Is there any other value you can assign to the substitution variable to solve this integral? $$\int \frac{x+3}{\sqrt{x+2}}dx$$ Substituting $u = x + 2$: $$du = dx; u +1 = x+3 ,$$ and we get this ...
-1
votes
1answer
50 views

How to solve $\int \frac{\ln{(x^4 + x^2)}}{x^2} \mathrm{d}x$? [closed]

$$\int \frac{\ln{(x^4 + x^2)}}{x^2} \mathrm{d}x$$ Can't solve this integral. I have been sitting over it for already an hour and still can't find an obvious solution. Please help.
2
votes
2answers
197 views

How to integrate $\frac{1}{(1 + x^5)(1 + x^7)}$

My cousin who is in high school asked me if it is possible to integrate $$ \int \frac{1}{(1 + x^5)(1 + x^7)} \, dx $$ I checked the list of integrals of rational functions on Wikipedia link and it ...
-2
votes
4answers
53 views

why $\int{\cos({\pi}t)} dt = \frac{1}{\pi}\sin({\pi}t)$?

Why? $$\int{\cos({\pi}t)} dt = \frac{1}{\pi}\sin({\pi}t)$$ the indefinite integral of $ \cos(x) = \sin(x) $ isn't it? so this suppose to be $$\int{\cos({\pi}t)} dt = \sin({\pi}t)$$ Why in the ...
1
vote
1answer
38 views

Integrals involving whittaker functions.

I want to compute the following integrals: $$ \int y^{a} e^{\frac{1}{2}y}M_{k,m}(y)dy $$ where a is an arbitrary constant and $M_{k,m}$ is a whittaker function of the first kind. I already know that ...