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Questions tagged [indefinite-integrals]

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

-1
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1answer
16 views

Bound on an indefinite integral in kernel density estimation

I am having trouble on what is probably a simple step in the proof of Theorem 24.1 in Asymptotic Statistics by Van der Vaart. Let $\int K(y) dy = 1$. The author writes: $$h^4 \int K(y)y^2 dy \int \...
5
votes
2answers
92 views

Finding $\int\lfloor x\rfloor\cdot |\sin(\pi x)| dx$

Find $$\int\lfloor x\rfloor\cdot |\sin(\pi x)| dx$$ Any help would be appreciated. The function is from $\mathbb{R}$ to $\mathbb{R}$, and so should be the result. The best I got thus far is: $$\...
1
vote
1answer
57 views

Integration of $f(x)= \exp(-ax-b\sin^{-1}(cx))$

Is there an analytical solution for the following integral? $$\int \exp(-ax-b\sin^{-1}(cx)) dx$$ Mathematica was unable to solve it, and I tried myself by integration by parts but without success.
3
votes
1answer
91 views

Integral of $ \int x^{n-1}W(x)dx $

How to prove that : $$ \int x^{n-1}W(x)dx = \frac {x^ne^{[-nW(x)]}[-nW(-x)]^{-n}[n\Gamma(n+1, -nW(x)- \Gamma(n+2, -nW(x))]} {n^2} $$ Where $W(x)$ is the Lambert-W function https://en.wikipedia.org/...
0
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0answers
16 views

What values of $a$ make $\frac{a-1}{3-a} x^{3-a} |_1^{\infty} - (\frac{a-1}{2-1})^2(x^{2-a} |_1^{\infty})^2$ finite?

What values of $a$ make $\frac{a-1}{3-a} x^{3-a} |_1^{\infty} - (\frac{a-1}{2-1})^2(x^{2-a} |_1^{\infty})^2$ finite? Is it a necessary and sufficient condition to simply make both the first and the ...
1
vote
1answer
40 views

integral of $\sec x \tan x$

method 1:Substitution $\int \sec x\tan x dx=\int \frac{\sin x}{\cos ^2x}dx$ Let $u=\cos x \implies -du=\sin x dx$ $-\int \frac{1}{u^2}du=\frac{1}{u}+c$ So $\int \sec x\tan x dx=\frac{1}{\cos x}=\...
1
vote
4answers
118 views

Any hints on how to compute this integral?

Could anyone please give me a hint on how to compute the following integral? $$\int \sqrt{\frac{x-2}{x^7}} \, \mathrm d x$$ I'm not required to use hyperbolic/ inverse trigonometric functions.
1
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2answers
81 views

$\int \frac{dx}{\sin^3{x}}$ possible with universal substitution?

If function is of the form $F(\sin{x},\cos{x})dx$, then I can apply $t = \tan{\frac{x}{2}}$ substition. Examples: $\int \frac{1}{4+3\cos{x}}dx$ $\int \frac{\sin{x}-1}{\cos{x}+2}dx$ $\int \frac{\...
0
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0answers
37 views

Calculating value of Independently using Integral [duplicate]

Let $$I(a)= \int_0^\infty \frac{dx}{(1+x^a)(1+x^2)}$$ where $a\in\mathbb{R}$. Evaluate $I(a)$ and show that its value is independent of $a.$
4
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1answer
272 views

Proving that $\int x^{x^{x^{.^{…}}}} dx= \sum_{n=1}^{\infty}\frac {(-n)^{n-1}}{n!} \Gamma(n, -\ln x)$ [Proof Verification]

Please check if I solved this correctly and if there are any mistakes. Many of steps are well known properties , so I might have skipped them. To prove: $$\int x^{x^{x^{.^{.......}}}} dx= - \sum_{n=...
1
vote
3answers
64 views

$\int \cos^4{x}dx$ unsolvable with $t = \tan{x}$?

I have been told that "universal substitution always works", so I wanted to give it a try on this specific integral. $\int \cos^4{x}dx$ For some reason it does not work. Please note that I solved ...
1
vote
1answer
63 views

Solving $\int \frac{\sqrt{(x-2)^3}}{(\sqrt{x+1})^2} dx$

This is the last indefinite integral I am attempting before proceeding to definite integrals study. Any ideas how to solve it? $$\int \frac{\sqrt{(x-2)^3}}{(\sqrt{x+1})^2} dx$$ My attempt: (not the ...
1
vote
1answer
64 views

Is there any other method of integration? besides the best known .

Is there any other method of integration besides the best known as: Substitution, Integration by parts, Trigonometric, Trigonometric Substitutions, Partial Fractions, Improper Integrals, and ...
0
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1answer
49 views

Integral with two distinct roots in them, such as: $\int \frac{\sqrt{x}}{x^2(\sqrt{x+1}+\sqrt{x})}dx$

I'm getting familiar with basic indefinite integrals and these are the hardest ones I've met so far: $\int \frac{\sqrt{x}}{x^2(\sqrt{x+1}+\sqrt{x})}dx$ $\int \frac{\sqrt[3]{x+2}-\sqrt[3]{x}}{x^2(\...
0
votes
1answer
73 views

Is there a calculus attempt at this question:

This question is based on the electric field. This is the question which I would like to solve using this integral: This is the equation for Electric Field. $$E=\int \frac{dq}{r^s}$$ I was wondering ...
0
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0answers
30 views

Help in applying the boundaries of an elliptic integral

I need urgent help in determining the final result of the definite integral: $$\int_0^{2\pi}\left (b\cos(x) + \sqrt{a^2 - b^2\sin^2(x)}\right)\, \mathrm{d}x.$$ I know, by using an online integral ...
-2
votes
1answer
104 views

Evaluate $\int \frac {e^x} {\sqrt {e^x + e^{2x}}} \text {dx}.$ [closed]

I started out with $U$-sub, making $e^x$ as $U.$ Replaced everything with $U$ and now I am stuck!
3
votes
2answers
145 views

Shortcut for value of $f(1)$ where $f(x) = \int e^x \left(\arctan x + \frac {2x}{(1+x^2)^2}\right)\,dx$

If $$f(x) = \int e^x \biggr(\arctan x + \frac {2x}{(1+x^2)^2}\biggr)\,dx$$ and $f(0)=0$ then value of $f(1)$ is? This is actually a Joint Entrance Examination question so I have to do it in two ...
3
votes
2answers
68 views

Irrational integral $\int \frac{1}{\sqrt{x}} \sqrt{\frac{\sqrt{x}-2}{\sqrt{x}+2}}dx$

Would anyone be able to verify if this integral is calculated correctly? $$\int \frac{1}{\sqrt{x}} \sqrt{\frac{\sqrt{x}-2}{\sqrt{x}+2}}dx$$ My attempt: substitute:$\left(t = \sqrt{x}, t^2 = x, ...
2
votes
1answer
63 views

Calculating $\int \frac{\sqrt{\sqrt[3]{x} - 2}}{x}dx$

Can someone help me calculate this integral? $\int \frac{\sqrt{\sqrt[3]{x} - 2}}{x}dx$ I tried this substitution: $\Bigg(t = \sqrt[3]{x}, t^3 = x, 3t^2dt=dx\Bigg)$ which reduces the integral to: $...
1
vote
1answer
30 views

How can I find $\int \frac{1}{\sqrt{x^2 + x + 1}} dx$ [duplicate]

My question is, how can I evaluate the following integral? Am I supposed to use Euler substitution here or is there a simpler way? $$\int \frac{1}{\sqrt{x^2 + x + 1}} dx$$ Thanks.
1
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1answer
67 views

Does my book lie about $\int \frac{\arccos(\frac{x}{2})}{\sqrt{4-x^2}}dx$?

I dropped out of university for 3 years or so and now I am want to come back so I started solving some calculus assignments in order to wash off my rusty knowledge of it. Anyway, as you can probably ...
1
vote
2answers
72 views

Evaluate $\int \frac{1}{2+\sin x+\cos x}dx.$ [duplicate]

$$\int \frac{1}{2+\sin x+\cos x}dx.$$ My attempts: Let $y = \sin x+\cos x.\implies \frac{dy}{dx}=\cos x-\sin x=y'.$ $$\int \frac{1}{2+\sin x+\cos x}dx=\int\frac{1}{2+y}\frac{dy}{y'}.$$ And I tried ...
0
votes
1answer
76 views

trying to solve an integral

Im trying to evaluate the following integral $$ \frac{1}{4 \pi \sqrt{t} } \int\limits_{-\infty}^{\infty} \exp\left( - \frac{(y+2)^2 }{4} - \frac{(x-y)^2 }{4t} \right) - \exp\left( - \frac{(y-2)^2 }{4}...
0
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1answer
29 views

Trigonometric integrals involving tangent

So I came across a problem after answering the integral. The problem was: $\int\tan^3(3x)dx$. This is to be integrated. This is how I did it: $$\begin{align}\int\tan^2(3x)\tan(3x)dx&=\int(\sec^2-...
0
votes
2answers
60 views

Integrate $\int \frac{\cos^2 x}{(1 - \cos x)\sin x} dx$

Integrate $\int \frac{\cos^2{x}}{(1 - \cos{x})\sin x} dx$ So far I have gotten here: $-\int -\frac{\cos^2 (x) \sin x}{(\cos (x) - 1)(\cos^2 (x)-1)} dx$ here I can substitute $u = \cos x$ , $ -\...
3
votes
2answers
32 views

Evaluate $\int \frac{ 2\exp\left((-\tan^2(t))/a^2\right) }{\cos^3(t)a^2}dt$ using substitution.

Evaluate : $\displaystyle\int \frac{ 2\exp\left((-\tan^2(t))/a^2\right) }{\cos^3(t)a^2}dt$ The hint was to use $x=\cos(t)$ and the fact that $\int f'(x)e^f(x)dx=e^f(x)$. Since $$x=\cos(t)\;\text{...
0
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0answers
25 views

Bessel function limit

I was doing a physics problem and encountered with following limit of Bessel function : $\lim_{R\to\infty} R^2J_n(\lambda R)$ and $\lambda = \sqrt{\omega^2 - \frac{1}{R^2}}$ I got this limit from ...
2
votes
3answers
200 views

Integrate $\int\frac{\sin^{-1} (x)}{(1-x^2)^{3/4}} \,\mathrm d x$

Integrate $$\int\frac{\sin^{-1} (x)}{(1-x^2)^{\frac{3}{4}}} \,\mathrm d x$$ I have followed some steps from here, but am not able to solve this question. Any help would be appreciated. Update: After ...
0
votes
1answer
36 views

Partial fraction decomposition with two variables

While I was solving some exercises as training for a test I'm gonna have, I've noticed that some integrals that I have to solve, I don't know how to do them, and I have no explanation on my papers and ...
2
votes
2answers
71 views

Ways to simplify arctan() in integral results?

Lately when I was computing $$\int\frac{\mathrm{d}x}{1+\sqrt{1-2x-x^2}},$$I got the result$$-2\arctan{\frac{\sqrt{1-2x-x^2}-1}{x}}-\ln\left(1-\frac{\sqrt{1-2x-x^2}-1}{x}\right)+\mathbf{C}.$$However, ...
7
votes
2answers
137 views

Is $e^{\int \frac {1}{x}dx}$ equal to $x$ or $|x|$?

I encountered this expression quite a lot of times as a part of the integrating factor while solving linear differential equations. $$e^{\int \frac {1}{x}dx}$$ For sometime, I wrote it as $x$, and ...
0
votes
2answers
43 views

Integrate $\int \frac{\sqrt{9x^2-1}}{2x}dx$

What is $\int \frac{\sqrt{9x^2-1}}{2x}dx$? I tried to form a triangle with $\cos\theta=\frac{1}{3x}$ and $\sin\theta=\frac{\sqrt{9x^2-1}}{3x}$ to use as substitution. But I can't get rid of all the $...
0
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0answers
35 views

How to proof $\int_{-\infty}^{+\infty}\sin(w_1*t)\sin(w_2*t)\,dt = 0$ if $w_1 \neq w_2$

In my math script (signal theory) it says that two functions are orthogonal to each other when $\int_{-\infty}^{+\infty}s^\star(t)u(t)\,dt = 0$. Now I want to prove that $$\int_{-\infty}^{+\infty}\sin(...
0
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0answers
42 views

What is the default definition of $F(x)$ given $f(x)$?

To avoid the confusing (to me) notation, let's call $g(x)$ the antiderivative of $f(x)$, such that $g'(x) = f(x)$. I've seen $F(c)$ defined as the definite integral of $f(x)$ from $0$ to $c$ with ...
4
votes
1answer
79 views

Is this integral unsolvable?

So I took an integration test in AP Calculus yesterday and everything went smoothly except for one question. $$\int \frac{e^x}{x^2}dx$$ I tried chain rule, $u$ substitution, and all methods we have ...
1
vote
2answers
55 views

Indefinite integral $\int \frac{(1-x)^3}{x \sqrt[3]{x}}$

Find the indefinite integral $\int \frac{(1-x)^3}{x \sqrt[3]{x}}$: I guess the fay forward would be to find a suitable substitution, but I am struggling with that.
1
vote
3answers
42 views

Simple integral with $e^x$ - how to decompose it?

How to calculate this integral? $$\int \frac{e^{2x}+2}{e^x+1}dx$$ I have tried various substitions such as: $t = e^x, t = e^x + 1, t = e^x +2, t = e^{2x}$ and none seem to work. According to ...
0
votes
1answer
68 views

Anti-derivative of $\frac{\exp(x)-1}{x}$

I am looking for the antiderivative of $$\frac{\exp(x)-1}{x}$$ I showed that it is equivalent to calculate $$\sum_{n=1}^{\infty}\frac1n \frac{x^n}{n!}$$ but I can't find both of the solutions. If ...
0
votes
2answers
38 views

Indefinite integral of rational polynomial function

How do I integrate $\int \frac {(x-3)}{(x^2-2x+4)^2} dx $ I know that I have to integrate via recursion. But doing so I get to a point where I have to find $\int \frac {x^2}{(x^2-2x+4)^2} dx $ which ...
0
votes
1answer
55 views

Calculating an infinite integral of log-normal distribution

The integral is: $\int^\infty_0 x \exp{\Big(\frac{-(\log{x}-\mu)^2}{2\sigma^2}\Big)}dx \ \ \ $ (it is a second moment of log-normal distribution). I've tried several subsitutions, such as $u=\log{...
0
votes
1answer
30 views

Primitive of a composite function

I'm reading Zorich, Mathematical Analysis I, and I found a not clear step in the paragraph on Primitives. The particular sentence is shown below (adapted). From the definition of primitive of a ...
1
vote
3answers
42 views

Finding $\int\frac{\sin x+\tan x}{\cos x+\csc x}dx$

Finding $\displaystyle \int\frac{\sin x+\tan x}{\cos x+\csc x}dx$ what i try $\displaystyle \Lambda =\int\frac{\sin^2 x(1+\cos x)}{\cos x(\sin x\cos x+1)}dx$ $\displaystyle \Lambda=\int\frac{\sin^4 ...
1
vote
2answers
53 views

Can some improper integrals be directly evaluated?

Consider the following improper integral: $\displaystyle \lim\limits_{\delta{r} \to 0} \int^r_{\delta r} f(r) dr \tag{1}$ where $f(r)$ is finite everywhere and $f(0)=$ not defined. Then, will ...
4
votes
1answer
93 views

$\int\frac{dx}{x(x+1)(x+2)\cdot\space…\space\cdot(x+n)}$ [duplicate]

I've been trying to solve explicitly the following indefinite integral: $$\int\frac{dx}{x(x+1)(x+2)\cdot\space...\space\cdot(x+n)}$$ I tried to perform partial fraction decomposition, and after ...
1
vote
2answers
75 views

Calculate the integral $\int \frac{2-3x}{2+3x} \sqrt{\frac{1+x}{1-x}}dx$ [duplicate]

I have to calculate the integral $\int \frac{2-3x}{2+3x} \sqrt{\frac{1+x}{1-x}}dx$. I tried the following substitutions: $x \rightarrow \frac{1+t}{1-t}, x \rightarrow \frac{1-t}{1+t}, x \rightarrow \...
0
votes
0answers
35 views

How to integrate Heaviside function multiplied with a function

For example Wolfram|Alpha gave me this result. But I can't understand how do we achieve this result. How do we take indefinite integral of a function multiplied by heaviside function ?
0
votes
2answers
38 views

Any simple integration to this indefinite integral?

$I =\displaystyle\int \dfrac{\sqrt{4+9x^4}}{x^3}dx$ One method we have tried is to use the substitution $x^2=\displaystyle\frac2{3\tan\theta}$ ,but it seems hard to change back the $\theta$ to x in ...
4
votes
1answer
52 views

Find $\int |\sin(x) + \cos(x)|\ dx$

$$\int |\sin(x) + \cos(x)|\ dx$$ Do I just do: $$\operatorname{sgn}(\sin(x) + \cos (x)) \int \sin(x) + \cos(x) \ dx = \frac{\sin(x) + \cos (x)}{|\sin(x) + \cos(x)|} \int \sin(x) + \cos(x)\ dx$$ ...
1
vote
2answers
44 views

Find $\int \frac {1} {(x-a)^n} dx$

Find $\int \frac {1} {(x-a)^n} dx$ where $n \in \mathbb{N}, a \in \mathbb{R}$ Am I supposed to solve this using substitution?