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

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0
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0answers
38 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
25 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
23 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
55 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
86 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
19 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
votes
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
21 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
45 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
15 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
41 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 ...
1
vote
4answers
37 views

What would the value of this integral be when I apply these conditions? (My answer appears wrong for some reason) [on hold]

$$\int\frac{1}{(x+a)(x+b)}\,dx = \left(\frac{\ln|x+a|}{b-a}\right)+\left (\frac{\ln|x+b|}{a-b}\right)\,$$ What would the value of this expression be when $a \ne b$, then what would the value of ...
2
votes
2answers
62 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
30 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
47 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 - ...
-5
votes
1answer
66 views

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

Could someone please show me how to integrate $1+\ln x$?
4
votes
4answers
64 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
52 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
52 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 ...
0
votes
1answer
48 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
192 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 ...
-1
votes
4answers
52 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
37 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
55 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
89 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 ...
3
votes
0answers
38 views

Partial Fractions Integration Help [closed]

So this is the integral and i have tried going through it many times and i can't seem to figure out where i went wrong. The answer in the box is the final answer i came up with. I've used up all my ...
2
votes
1answer
38 views

How do I solve $∫4cos^2(x) dx$? [duplicate]

I have the basic idea of how to work out the integral of a trig function, but am having trouble in applying the concept. Would really appreciate it if someone could help me. Thanks!
0
votes
1answer
46 views

Integration - Evaluating the integral $ \int\sqrt{r^2-x^2}\,{\rm d}x $

I am having some trouble with the problem below. I cannot see where the $r^2$ comes from on the fourth line of the answer below. I have the integral $$ \int\sqrt{r^2-x^2}\,{\rm d}x $$ I make the ...
2
votes
3answers
32 views

Convergence test of $S=\frac{1}{\ln 2} \sum_{k=1}^\infty \ln (1+\frac{1}{k(k+2)}) \ln k$

Does S converge? (The answer says it converges) $S=\frac{1}{\ln 2} \sum_{k=1}^\infty \ln (1+\frac{1}{k(k+2)}) \ln k$ My attempt: Comparison test: $\ln (1+\frac{1}{k(k+2)}) \ln k \lt \ln 2 ...
0
votes
0answers
51 views

What substitution can work in $\int (x^2 \sqrt{x^2 + 1})e^{x (1 - \log{x})} dx$

$$\int (x^2 \sqrt{x^2 + 1})e^{x (1 - \log{x})} dx$$ I tried $t =x^2 + 1$, $t = \sqrt{x^2 + 1}$ and $t = x (1 - \log{x})$, but these don't seem to work. Any ideas?
1
vote
4answers
54 views

Integration by parts - hint

I'm stuck on a passage on my textbook: $$ \int \frac{1}{(1+t^2)^3} dt = \frac{t}{4(t^2+1)^2}+\frac{3}{4} \int \frac{1}{(t^2+1)^2} dt$$ I know that it should be easy but I just can't figure out what ...
-1
votes
0answers
74 views

Compute $\int_{0}^{1}\frac{x^3-x^2}{\ln(x)}\,{\rm d}x$

Compute $$\int_{0}^{1}\frac{x^3-x^2}{\ln(x)}\,{\rm d}x$$ I tried dividing it into 2 different integrals but I found that none of them converges, I'm pretty much stuck now.
1
vote
4answers
45 views

$\int \frac{2x}{9x^2+3}dx=?$

So this one seems very easy. And it really is (I guess). But I have an issue with this one. I solved it this way: \begin{align} & \int \frac{2x}{9x^2+3}dx=\frac{1}{6} \int ...
0
votes
3answers
38 views

Help reaching the solution to the integral

I have the following equation: $$ \frac{dR}{dt}=k_{o}(1-R)-k_{a}R $$ and I want to integrate the equation, whose solution is: $$ ...
2
votes
4answers
78 views

How to compute the integral of $ \frac{\sqrt{(x^2+1)}}{x^2}$?

II have spent hours trying to find the right substitution but I have had no results. I have tried using $x=\tan u$ but it did not give the result shown in my book.
3
votes
3answers
75 views

Evaluate $\int \frac{2-3x}{2+3x}\sqrt\frac{1+x}{1-x}dx$

Evaluate $$\int \frac{2-3x}{2+3x}\sqrt\frac{1+x}{1-x}d x$$ What substitution should I use ? $\sqrt\frac{1+x}{1-x}$ suggests $x=cos2\theta $ but its not useful in $\frac{2-3x}{2+3x}$
0
votes
2answers
42 views

Prove that the integral of $\frac{dx}{\sqrt{x-x^2}}$ is equal to $2\sin^{-1}\sqrt x+C$ using $u = \sqrt{x}$. [closed]

I know that you have to change $u=\sqrt x $ to $u^2 = x$. But I don't know what to do next?
0
votes
2answers
49 views

Need some hints to solve $ \int \frac 1 { \sqrt{x+\sqrt{x^3}} } dx $ (from $ \int \frac {x} { \sqrt{ 1 + x^2+\sqrt{(1 + x^2)^3}} } dx $)

I'm working on the indefinite integral for $ \int \frac {x} { \sqrt{ 1 + x^2+\sqrt{(1 + x^2)^3}} } dx $ and after u-substitution with $ u=1+x^2 $ and $ \frac {du}{dx}=2x $ I get $ \int \frac 1 { ...
2
votes
3answers
72 views

How do you find the indefinite integral for $\int \frac1{(x^2+1)^{3/2}} dx $?

How do you find the indefinite integral for $\int \frac1{(x^2+1)^{3/2}} dx $ ? According to the Apostol's Calculus Vol1 textbook this can be done by integration by substitution but not sure how.
0
votes
1answer
47 views

when $\int x^m (a+bx^n)^p dx$ is elementary function

In one of the answers of the question Integration of sqrt Sin x dx, I saw something similiar to that: $m,n,p \neq 0 \in \mathbb{Q}$ $\int x^m (a+bx^n)^p dx$ is elementary function $\implies$ ...
0
votes
2answers
47 views

Calculate the indefinite integral $\int \frac{dx}{({x^2-2x+5})^\frac{3}{2}} $

I have this. $$\int \frac{dx}{({x^2-2x+5})^\frac{3}{2}} $$ I tried to define - $ u = x-1 $ $\int \frac{du}{8(\frac{u^2}{4}+1)^{\frac{3}{2}}} = \frac{1}{8}\int ...
2
votes
2answers
78 views

Integral involving power of logarithm

I was wondering if we can compute the following integral: $$ I = \int_{0}^{1}{e^{\alpha y} y^{\beta} (\ln y)^m {\rm d}y} $$ where $m \in \mathbb{N}$, $\alpha > 0$, $\beta>0$.
2
votes
3answers
95 views

Integrating $\int \frac{u \,du}{(a^2+u^2)^{3/2}}$

How does one integrate $$\int \frac{u \,du}{(a^2+u^2)^{3/2}} ?$$ Looking at it, the substitution rule seems like method of choice. What is the strategy here for choosing a substitution?
3
votes
2answers
69 views

Compute this integral (Is there a trick hidden to make it eassier?)

I need some tips to compute this integral: $$ \int\,\dfrac{\sqrt{x^2-1}}{x^5\sqrt{9x^2-1}}\,dx $$ What I did was express the denominator in the following form: $$ ...