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Questions tagged [indefinite-integrals]

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

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38 views

Inequality with extreme values

$f:[0,1]\rightarrow \mathbb{R}$ is continuous and non-constant function. $F$ is the indefinite integral of $f$ such that $F(0)=F(1)=0$. $m$ and $M$ are the minimum and the maximum values of $f$. Now ...
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0answers
32 views

How to evaluate this integral with exponent of an exponent?

I have the following integral which I need to evaluate but don't even know where to begin other than knowing I need to use u-substitution: $$\int_1^\sqrt{3}2x^{x^{2}}$$ So far I know that $u=x^{2}$ ...
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15 views

how to break limits of an integral of joint variables [on hold]

if x1 & x2 are uniform random variables then find pdf of Z= X1+ X2 f(x1,x2) = 2,, where 0 <- x1 <- x2 <- 1 BY using CDF method find pdf for Z
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1answer
59 views

Solving $\iint \frac{1}{(x^2+y^2+1)^{3/2}} dx dy $

This is a question in a book of statistics and probability. To prove that this function is a Probability density function, we should solve it to get the answer equals to 1. I haven't had to deal with ...
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21 views

Non-product measure proof of Fubini’s theorem [on hold]

Can you state/direct me to a proof of Fubini’s theorem that does not rely on product measure ?
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4answers
62 views

Evaluate the indefinite integral $\int {(1-x^2)^{-3/2}}dx$

Evaluate the indefinite integral $$\int \frac {dx}{(1-x^2)^{3/2}} .$$ My answer: I have taken $u = 1-x^2$ but I arrived at the integral of $\frac{-1}{(u^3 - u^4)^{1/2}}$. What should I do next?
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2answers
24 views

Are these legitimate rules/formula for integration without using the substitution method?

I'm talking about: $\int(ax+b)^ndx=\frac{(ax+b)^{n+1}}{(n+1)(a)}$ $\int\frac{1}{ax+b}dx=\frac{1}{a}ln(ax+b)$ $\int e^{ax+b}dx=\frac{e^{ax+b}}{a}$ $\int a^{ax+b}dx=\frac{a^{ax+b}}{(ln|a|)(a)}$ Of ...
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1answer
34 views

Reduction formula for $\int\frac{dx}{(ax^2+b)^n}$

I recently stumbled upon the following reduction formula on the internet which I am so far unable to prove. $$I_n=\int\frac{\mathrm{d}x}{(ax^2+b)^n}\\I_n=\frac{x}{2b(n-1)(ax^2+b)^{n-1}}+\frac{2n-3}{2b(...
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0answers
27 views

Find the analytic form of expression for the below integral

$$ \int_{0}^{\infty} \frac{1}{a \hspace{0.05cm} e^{br} + b \hspace{0.05cm} e^{ar} + c \hspace{0.05cm} e^{(a+b-c)r} + d \hspace{0.05cm} e^{(a+b-d)r}} dr \hspace{0.1cm}; \hspace{0.9cm} ...
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2answers
26 views

$\int\limits^\infty_{-\infty} xe^{-|(x-u)|} dx = ?$

I'm trying to solve this integral with absolute values. Wolframalpha shows that $\int\limits^\infty_{-\infty} xe^{-|(x-u)|} dx = 2u$, but when I break the absolute value into two integrals I don't get ...
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2answers
37 views

Questions about Indefinite Integrals and U - Substitution

Let me begin with an example. If we were to integrate the indefinite integral of $(2x)^2$ with respect to $x$ with u-substitution, we would first say that $u=2x$ and therefore $du=2dx$. In order to ...
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0answers
23 views

how to estimate a multiple indefinite integral with constant integrand?

Let $\rho_i,\eta_i\in[a,b]$. How to prove the following inequality? \begin{equation*} \left|\int_{\eta_{k+1}}^{\rho_{k+1}}d\rho_k\int_{\eta_{k}}^{\rho_{k}}d\rho_{k-1}\cdots\int_{\eta_{2}}^{\rho_2}d\...
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1answer
64 views

Find $f$ such that : $f$ is absolutely integrable, $f'$ is absolutely integrable and such that $f$ is not $1/2$-Hölder

I am trying to find a function $f: \mathbb{R}^+ \to \mathbb{R}^+$ that fullfils the following conditions $$f \in \mathcal{C}^1(\mathbb{R}^+,\mathbb{R}^+)$$ $$\int_{\mathbb{R}^+} f \in \...
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1answer
61 views

How to integrate $\int_{1/2}^{2}\frac{\sin(\frac{x^{2}-1}{x})}{x} dx$

How to integrate $$\int_{1/2}^{2}\frac{\sin(\frac{x^{2}-1}{x})}{x} dx$$ I tried a couple of thing such as integration by parts by letting $\frac{1}{x}$ as an intigrable function but got $$-\int_{1/...
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15 views

How do I put $g(x,y)$ into the second integration in $\int_{-\infty}^\infty g(x,y)(\int_{-\infty}^\infty f(x,y)\mathop{\mathrm dy})\mathrm dx $

We have $$\int_{-\infty}^\infty g(x,y)\left(\int_{-\infty}^\infty f(x,y)\mathop{\mathrm dy}\right)\mathrm dx $$ What I would like to have is to put $g(x,y)$ into the second integral, so that it is ...
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96 views

Any way to solve this integral?

I have not been able to solve the integral, any idea of how to do it? $$\int \frac{t}{\sqrt[t]{e}+1}\text dt$$
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2answers
52 views

Different result with Mathematica for $\int \frac{x}{x^2 + \frac{1}{4}}\ dx$?

I am confused why Mathematica gives a different answer with this simple integral: $$ \int \frac{x}{x^2 + \frac{1}{4}}\ dx = \frac{1}{2}\log(x^2 +\frac{1}{4}) + C. $$ Mathematica produces $$ \...
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21 views

Fixing the bounds of an indefinite integral

Question: $$\frac{\partial^2 u}{\partial x^2}-\frac{\partial^2 u}{\partial t^2}=f(x,t) \qquad \qquad u(x,0)=\frac{\partial u}{\partial t}(x,0)=0$$ Solve this equation, writing the solution in the ...
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2answers
52 views

Integrating $\sec^5x$ using integration by parts

I'd like to know what I did wrong in my solution using this particular $u$ and $v$ $\int\sec^5x dx$ $u=\sec x$ $du=\sec x\tan x$ $v=\int \sec^4x dx = (\tan^3x)/3+\tan x$ After completing the ...
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1answer
175 views

For which values of $a$ is $f$ primitivable?

Let $a \in \mathbb{R}$ and $p,q$ be natural numbers with $p \geq q+2.$ For which values of $a$ is the function $$f(x) = \begin{cases} \frac{1}{x}\sin \frac{1}{x^p}\sin \frac{1}{x^q}, &x \neq 0 ...
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3answers
58 views

Questions about U - Substitution and Integration

I have two questions: I just learned about U-Substitution in class, and while I'm able to apply it, I'm a bit confused on some of the theory behind it. The thing that most confuses me is that if we ...
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4answers
57 views

Solving two integrals

Let $f(x) = x^{-2}e^{-x}$ and $g(x) = 2x^{-3}e^{-x}$ . Find $\int f(x)dx$ and $\int g(x)dx$ . I tried to use substitution and integration by parts but didn't help .
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1answer
63 views

How do I evaluate this integral by parts?

So before anyone asks yes this is a homework question but my professor has allowed us to use the internet for help. Anyways my professor would like us to evaluate this integral by parts. $$ \int t \...
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3answers
81 views

Integral of $\int \frac{-x^2+2x-3}{x^3-x^2+x-1}dx$

I have this simple integral: $\int \frac{-x^2+2x-3}{x^3-x^2+x-1}dx$ and I can't come up with the correct answer. Here's what I did: I found the roots for $x^3-x^2+x-1$ so I could do partial ...
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6answers
85 views

Any other way to solve $ \int\frac{1}{1+\tan x} \,dx $?

$$ \int\frac{1}{1+\tan x} \,dx $$ One method to solve is to use $ \tan{2 \theta} = \frac{2\tan{\theta}}{1-\tan^2{\theta}} $ Second is by reducing integrand in terms of $ \sin x $ and $\cos x$ ...
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1answer
21 views

Can you multiply an integral by f(x)/f(x) where deg(f(x))>0?

I was on instagram and someone posted this: https://www.instagram.com/p/BplPOrlAvob/ I haven't been on here in some time, so please excuse my poor formatting, but the integral to be solved is: $\int ...
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1answer
70 views

$\int\sqrt{1-\tan x}~\mathrm{d}{x}.$ (Integral of a trigonometric function under square root)

$\int\sqrt{1-\tan x}~\mathrm{d}{x}.$ is an integral which I am not able to solve. I have restricted my ideas on trigonometric substitution but cannot conclude to an answer...will really appreciate if ...
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1answer
67 views

Supposed method of integration: “long dividing” by $d$

I came across this seemingly interesting, but poorly exemplified, method of integrating: Since integration is the inverse of differentiation, you can think of integration as “dividing” by $d$. ...
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1answer
29 views

Given the density function, show that $c sin^{-1}(\sqrt{x})$ is the distribution function and find c.

I was able to find c which is $F(x) = c sin^{-1}(\sqrt{x})$ if $0 < x < 1$ $F(1) - F(0) $ $c (sin^{-1}(\sqrt{1})-sin^{-1}(\sqrt{0})) = 1$ $c (\frac{\pi}{2} - 0) = 1$ $c (\frac{\pi}{2}) = ...
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0answers
93 views

Equation in integral form

I've been working with this equation where the unknown factor is the function $f$ that can be complex: $$1 = f(\vec{x})\int_{\mathbb{R}^3}\ d^3y\ \frac{f(\vec{y})}{|\vec{x} - \vec{y}|^4}$$ Is there ...
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1answer
24 views

Finding the result of a definite integral when its limit is changed and integrand is not symmetrical

So from wikipedia (https://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions#Integrals_involving_only_exponential_functions), the an $x^2$ integral with a function of exponential raised ...
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0answers
23 views

Minimum distance between two functions involving integration

Let $f(x)$ and $g(x)$ are two differential functions such that $f(x) = \frac{x^3}{2} + 1 - x \int_0^x{g(t)}dt $ and $g(x) = x - \int_0^1{f(t)}dt $. Also suppose $\alpha = min |f(x) - g(x) |$ for all ...
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1answer
52 views

Integral in relative and center of mass variables

I'm trying to compute the following integral where $\vec{x}$ and $\vec{y}$ are integrated inside a sphere of radius $l$: $$I = \int\ d^3xd^3y\frac{1}{|\vec{x} - \vec{y}|^2}$$ To do it, I perform the ...
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1answer
44 views

Integral of polynomial times sine over $\mathbb{R}^+$

Computing the following integral I get: $$\int\ x^2\sin(a·x)\ dx = -\frac{x^2\cos(a·x)}{a} + \frac{2x\sin(a·x)}{a^2} + \frac{2\cos(a·x)}{a^3} \tag1$$ So, if I have to do this integral with the ...
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3answers
57 views

Find the indefinite integral of $\frac{1}{3\sin x + \sin^3 x }$ with respect to x [closed]

The required integral is: $\int \frac{1}{3\sin x + \sin^3 x }$ $dx$ I tried integrating this in many ways but never got the answer. Showing all the approaches will be pointless as there are several (...
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1answer
29 views

Is there a general pattern for all integrals of the form $\sin^{k/2}x$ where $k$ is an integer?

Though a CS student, my main hobby is mathematics, and was wondering about the primitives of rational powers of $\sin x$. I solved and found the primitive of the square root of $\sin x$ via a ...
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2answers
22 views

Can I get help solving this indefinite integral: (tan(x))^2*(sec(x))^1/2

I am quite stuck on how do I solve this indefinite integral, the integrand is: (tan(x))^2(sec(x))^1/2; I got it as a residue of a differential equation I was solving, I tried a lot of substitutions ...
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2answers
62 views

How to integrate $ \int\frac{x^3+|x|+1}{x^2+2|x|+1} $?

$$ \int\frac{x^3+|x|+1}{x^2+2|x|+1} $$ I tried redefining the function , but that could not help . I even tried making factors out of denominator which again proved futile . Any help to solve this ...
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3answers
112 views

How to integrate $\int{\frac{\ln(1+t)}{t^2+1}}dt$

I was given this problem, $$\int{\frac{x^2+1}{x^4-x^2+1}}\ln{(1+x-\frac{1}{x})}dx$$ Putting $x-\frac{1}{x}=t$, We get $$\int{\frac{\ln(1+t)}{1+t^2}}dt$$ But I am struggling to integrate after this ...
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2answers
61 views

Finding value of $ \int \frac{1}{x-\sqrt{9x^2+6x+4}}dx$

Finding value of $\displaystyle \int \frac{1}{x-\sqrt{9x^2+6x+4}}dx$ Let $$\displaystyle I = \int\frac{1}{x-\sqrt{9x^2+6x+4}}dx = \int\frac{x+\sqrt{9x^2+6x+4}}{-8x^2-6x-4}dx$$ $$I = -\frac{1}{8}\int ...
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1answer
45 views

Integration of $\frac{1}{\log(x)}$? [duplicate]

How to integrate $\frac{1}{\log(x)}$? I have tried integration by parts, but it is a never ending series with no specific general term. PS: It is indefinite integration.
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3answers
66 views

Prove $\int f(x)f'(x)\,dx = \frac {[f(x)]^2}2 + c$ through substitution

I've always taken integration by substitution for granted but recently I've learned that differentials can't fully be treated as variables and that the process of integration by substitution is really ...
2
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2answers
48 views

How to integrate $\int \frac{x-1}{x^2\ln(x)+x}dx$

According to Wolfram Alpha, $$\int \frac{x-1}{x^2\ln(x)+x}dx = \ln\left(\ln(x)+\frac{1}{x}\right)$$ but I don't really know how to do this.
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2answers
71 views

Is there any other method to solve $ \int \frac{1-\frac{1}{x^2}}{(x+\frac{1}{x})\cdot\sqrt{x^2+\frac{1}{x^2}}} $?

Question is $\newcommand{\dd}{\,\mathrm{d}}$ $$\int \frac{x^2 - 1}{(x^2+1)\cdot\sqrt{x^4+1}} \dd x$$ Solution given in book $$\int \frac{1-\frac{1}{x^2}}{\left(x+\frac{1}{x}\right)\cdot\sqrt{x^2+\...
2
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0answers
31 views

Extraneous Condition in the Hypothesis?

If $\{f_n\}$ is a sequence of continuously differentiable functions on $[a,b]$, $f_n \to f$ uniformly on $[a,b[$, and there is a function $g : [a,b] \to \Bbb{R}$ such that $f'_n \to g$ uniformly on $[...
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2answers
24 views

Integration by parts loop with coefficient

I leant that when the integral appears on the right side of the equation, it can be transferred accross to the left side, as in this post, but I'm trying to learn how to do this if there is a ...
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0answers
13 views

Indefinite Integral Solution Help

I stumbled across a problem I cannot find a solution to. Basically, I want to calculate $\int\frac{f(1-kx)}{x}dx$. F is a function of x. There is no information about the exact form of f, nor any ...
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2answers
85 views

How would you integrate $\frac{Si(x)}{x}$?

The function $Si(x)$ can be obtained when we integrate $\frac{\sin(x)}{x}$. But how would we go about integrating $\frac{Si(x)}{x}$? More information about the function $Si(x)$ can be found here ...
2
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1answer
53 views

How would you integrate the trignometric integral function Si(x)?

The function Si(x) can be obtained when we integrate $\frac {sin(x)}x$. But how would we go about integrating Si(x)? More information about the function Si(x) can be found here https://en.wikipedia....
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2answers
63 views

Integrate $\int x^x dx$

I have proceeded as follows: $$I = \int x^x dx = \int \sum \frac {(x \log x)^k}{k!} dx = \sum \frac {\Gamma [k+1, -(k+1) \log x]}{(-1)^k(k+1)^{k+1}k!} + C$$ But I am unable to go further to get rid of ...