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

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1
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1answer
56 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 ...
3
votes
4answers
67 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
51 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
62 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
40 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
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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
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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
71 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? $$\int\limits \frac{1}{\left(\cos^4(x) \cdot \sin^2(x)\right)}dx$$
-3
votes
0answers
69 views

the integral of exponential function [on hold]

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
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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
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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
53 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
vote
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
88 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
67 views

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

Could someone please show me how to integrate $1+\ln x$?
4
votes
4answers
66 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
54 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 ...
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votes
1answer
49 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 ...
1
vote
2answers
56 views

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

I successfully evaluated the following integral using partial fraction expansion, but am unsure of a few steps. $$ \int{\frac{x^2-2x-1}{(x-1)^2(x^2+1)}} = \int\left( \frac{A}{x-1} + \frac{B}{(x-1)^2} ...
2
votes
3answers
92 views

Integration by parts: $\int{\frac{dx}{(x^2 + a^2)^n}}$.

I need to show that the following holds using integration by parts: \begin{equation} \int{\frac{dx}{(x^2 + a^2)^n}} = \frac{x}{2a^2(n-1)(x^2 + a^2)^{n-1}} + \frac{2n - 3}{2a^2(n-1)} ...
0
votes
1answer
25 views

Quick question about trigonometric substitution $\int\frac{u+5}{u^2+9}du$

I'm trying to calculate this integral:: $$\int\frac{u+5}{u^2+9}du$$ which is a part of: $$\int\frac{e^x}{(e^x-5)(e^{2x}+9)}dx$$ The part I posted is the only one giving me a wrong answer. I am ...
1
vote
0answers
27 views

Calculus Net change and Indefinite Integrals

The marginal cost of producting $x$ tablet computers is $$C'(x) = 110 - 0.04 x + 0.0002\ x^2$$ dollars. What is the cost of producing $4000$ units if the setup cost is $70000$ Also If the ...