Questions tagged [indefinite-integrals]

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

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Alternative approach for $\int\frac{1}{\tan\left(\frac{1}{2}\csc^{-1} (x)\right) - \tan\left(\frac{1}{2}\sec^{-1}(x)\right)} \, dx$

This is example 2 in my "Integration Using Some Euler-Like Identities" blog post. $$\int\frac{1}{\tan\left(\frac{1}{2}\csc^{-1} (x)\right) - \tan\left(\frac{1}{2}\sec^{-1}(x)\right)} \, dx\...
Emmanuel José García's user avatar
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If I use this proposition to get an elementary primitive of $f$, I think $g(t)$ must be a monotonic function. ("Calculus" by Takeshi Saito)

The following is from "Calculus" (in Japanese) by Takeshi Saito. Proposition 4.3.9 Let $f(x)$ be a continuous function defined on an open interval $(a,b)$ and $g(t)$ be a continuously ...
佐武五郎's user avatar
2 votes
2 answers
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After integrating $\csc^2(x)\cot(x)$ why do I get $\cot^{2}(x) = \csc^{2}(x)$?

As the title may suggest, I come today with a question regarding a (seemingly) simple trigonometric integral, that being $$\int_{}^{} \csc^2(x)\cot(x) \,dx$$ I encountered this question in a Calculus ...
LogicBeDamned's user avatar
-1 votes
1 answer
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Solve integral of irrational function $\int \:\frac{2x^2-x+1}{\sqrt{2x^2+x-1}}dx$ [closed]

I've been practicing irrational functions and I've come to this one and I have been stuck, I'm unsure even where to begin and what method to use??
drafty nene's user avatar
-3 votes
1 answer
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Reduction formula $\int\frac{x^n}{1+x^2}dx$ [closed]

I have been trying to solve this problem but I'm stuck, any hints ? Suppose $I_n = \int\frac{x^n}{1+x^2}dx$. Use algebraic manipulation to show that $I_n= \frac{x^{n-1}}{n-1}- I_{n-2}$
Arqam Khalid's user avatar
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An indefinite integral within a definite integral and a confusion regarding positioning the variable.

So I have an integral to solve for (this comes out as the second term of integration by parts), which is supposed to give me a function of $a$. The integral is as follows: $\int_{\infty}^{a} 2 u\left (...
Caslone's user avatar
2 votes
1 answer
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Computing $\int_{0}^{\infty}\frac{x^{a}}{x^2 + x + 1}dx$ for $a\in (-1,1), a\neq 0$

I am trying to compute $$I = \int_{0}^{\infty}\frac{x^{a}}{x^2 + x + 1}dx$$ for $a\in (-1,1), a\neq 0$. I considered the contour integral $$\int_C \frac{z^{a}}{z^2 + z + 1}dz$$ where $C$ is the ...
idk31909310's user avatar
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2 answers
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The fastest solution for $\int \sqrt{x^2+1}\,dx$

The integral $(1)$ is typically attacked using trigonometric substitution, more specifically, using Case II, which involves substituting $x=a\tan{\theta}$. This method leads to having to evaluate the ...
Emmanuel José García's user avatar
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Calculating Total Gravitational Torque for Variable Density Stick with infinte length. [closed]

Question on Math Stack Exchange How to Calculate dτ/dx0 for Torque of a Stick with Variable Density? I am trying to solve a physics problem involving a stick of variable density ρ(x) and ...
M_Ahsan's user avatar
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1 answer
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Integration by parts of a Gaussian function.

I've attempted the following question which involves integration by parts. I've only recently learnt come across this skill, which means there is a very large chance that I'm doing something wrong. ...
Developer's user avatar
1 vote
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Any good books/sources about multiple indefinite integrals?

I want to learn some stuff about double, triple, ect. indefinite integrals, meaning antiderivatives of functions with multiple variables, but, I didn't find much online and I was wandering if any of ...
Mitsos YT's user avatar
4 votes
2 answers
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Does $ \int_{-\infty}^\infty \int_0^x e^{y^2/2} \, dy \; e^{-x^2/2} \, dx$ exist?

I am trying to figure out if the following integral exists $$ \int_{-\infty}^\infty \int_0^x e^{y^2/2} \, dy \; e^{-x^2/2} \, dx $$ For higher potentials in the exponent of the exponential, say $x^4$, ...
TrippyMushroom95's user avatar
2 votes
2 answers
71 views

Evaluate $\int{\frac{1}{x^3}}$ using u-sub.

I tried solving $\int{\frac{1}{x^3}}$ using u-sub instead of power rule and I got $-\frac{1}{2}x^{-2}$ instead of $\frac{x^4}{4}$. Its very possible I've made a very simple mistake or their is ...
Robert Barnett's user avatar
3 votes
2 answers
100 views

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

My attempt: Factor the denominator: The denominator can be factored as $(x + 1)(x + 3)$. Substitution: Let's substitute $u = x + 1$. Notice that $du = dx$. Rewrite the integral: Substitute $x$ and $dx$...
Nsnansn Jwjwj's user avatar
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Differential Equation on the vertical movement of a Slinky

I'm curious about solving a differential equation for the displacement vs. time on the vertical movement of a slinky when it is suspended from rest and starts to oscillate. I have gotten to these ...
QuantumYitian's user avatar
0 votes
1 answer
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Proof that a specific exponential integral converges (Admissibility of complex Morlet wavelet)

As part of a proof of the admissibility of the complex Morlet wavelet, I am trying to show that the following integral is positive and finite $$ 0<\int_0^\infty{\frac{(e^{\sigma \omega}-1)^2e^{-\...
Isaac Mortiboy's user avatar
1 vote
1 answer
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Mathematical Meaning of Antiderivatives

I'm largely a self-taught highschooler in basic Calculus and I'm utterly confused regarding what Indefinite integrals (or antiderivatives) do mean geometrically (if they really do), physically or ...
BlackKnight23's user avatar
3 votes
1 answer
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Solve the IVP $y'+2y = \frac{1}{1+x^2}$

Find the solution of the DE $$y'+2y = \frac{1}{1+x^2}\,\,\,\,\,\,\forall x \in \mathbb R$$ satisfying $y(0) = a$ where $a \in \mathbb R$ is a constant. My attempt: Since it's a linear ODE, therefore ...
Ark's user avatar
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4 votes
1 answer
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Prove that sum of integrals $= n$ for argument $n \in \mathbb{N}_{>1}$

ORIGINAL QUESTION (UPDATED): I have a function $f:\mathbb{R} \rightarrow \mathbb{R}$ containing an integral that involves the floor function: $$f(x):= - \lfloor x \rfloor \int_1^x \lfloor t \rfloor x \...
Richard Burke-Ward's user avatar
9 votes
4 answers
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How to prove the two answers to an integral are equivalent

I'm trying to do the integral: $$\int{\frac{1}{\sqrt{e^{-2x}-1}}}dx$$ So I try two ways to do it, the first method I used is to multiply $e^x$ on both sides first. $$\int{\frac{1}{\sqrt{e^{-2x}-1}}}dx$...
ACgroup's user avatar
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2 answers
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Integrate : $\int (\sin(x))^{2a-1} dx$

Question : Integrate $$\int (\sin x)^{2a-1} dx$$ where $a \in \mathbb{N}$. My Attempt : Using IBP on $$I_j=\int (\sin x)^{2j-2}\sin x dx= -(\sin x)^{2j-2}\cos x+\int(2j-2)(\sin x)^{2j-3}(\cos x)^2dx$$ ...
sparrow_2764's user avatar
5 votes
1 answer
250 views

Integrate : $\int e^{x+e^{x+e^x}}\, dx$

Question : $$\int e^{x+e^{x+e^x}} dx$$ source : Integration Techniques and Tricks University of Miami Mathematics Union Trevor Birenbaum My attempt: can be rewritten as $$\int e^x (e^{e^x})^{(e^{e^x})...
sparrow_2764's user avatar
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39 views

A indefinite integral question

The integral is that $$ \int\sqrt{ln\frac{(x+\sqrt{1+x^2})}{1+x^2}}\mathrm{d}x $$I have two questions. The first is that how can I think of that trick:$$\frac{1}{\sqrt{1+x^2}}\mathrm{d}x = \mathrm{d}\...
fishman's user avatar
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What is the value of the integral F(a)?

Given that: $$\int_a^T f(x) \,dx = F(T)$$ Doesn't this mean $F(a) = 0$, because I found the equation: $$\int_a^b f(x)\,dx = \left.F(x)\right\rvert_a^b = F(b)-F(a) $$ and I wondered what's the point ...
zizaaooo's user avatar
1 vote
1 answer
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What is the simplest way to approach this integral problem? To me, this problem seemed to necessitate some kind of gimmick.

How can I tackle this problem with as few abstract ideas as possible when attempting to 'purely' integrate from left to right? I've tried a few different versions of this term, but I'm not sure how to ...
Wayferer Alpha's user avatar
1 vote
0 answers
89 views

How to complete this proof of $\int_{0}^ \infty \frac{nx \arctan(x)}{(1+x)(n^2+x^2)}dx =\frac{\pi^2}{4}$?

I saw this problem:$\int_{0}^ \infty \frac{nx \arctan(x)}{(1+x)(n^2+x^2)}dx =\frac{\pi^2}{4}$ and I tried to solve it. Here is my attempt $\textbf{Claim: }$ If $f$ is continuous at $[0,1]$ then $\...
pie's user avatar
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2 votes
4 answers
158 views

How to integrate $\int \frac{3x^{4}+5x^{3}+7x^{2}+2x+3}{(x-6)^{5}}dx$?

Q) How to Integrate $\int \frac{3x^{4}+5x^{3}+7x^{2}+2x+3}{(x-6)^{5}}dx$ ? First of all let me tell what I think about this question. In my Coaching Institute, the chapter 'Integration' is over. This ...
Dropper's user avatar
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1 answer
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$\int \frac{\sqrt{a^2-x^2}}{x^2}dx$ using trigonometric substitition

I'm aware this question is more easily done using substitution by parts or euler substitution, but this was under a section in my book where we were asked to use trigonemtric substations. Substituting ...
Kryptic Coconut's user avatar
1 vote
0 answers
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Textbook says to integrate a fraction using 'Taylor's formula'?

I don't understand the solution my textbook gives for this problem: $$ \int \! \frac{x^3}{(x+1)^5} \, \mathrm{d}x $$ I thought it had to be done with partial fractions, but I couldn't get it right, ...
12_nihil_12's user avatar
1 vote
3 answers
221 views

Why do we call integration "accumulation of change"?

So in virtually all English-language calculus classes I have seen, we define integration as the "accumulation of change". And that makes sense to me intuitively, but when I think about it, I ...
software_dev_wannabe's user avatar
2 votes
0 answers
81 views

Approach to evaluate the integral $\int \frac{1}{{x^3}\sqrt{x^2-a^2}}dx$ using basic substitution.

So I was attempting a basic indefinite integral problem which goes like $$\int \frac{1}{{x^3}\sqrt{x^2-a^2}}dx$$ for which I used the substitution $x=a\sec\theta$ which on differentiating gave $$dx=a\...
Aayush's user avatar
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-2 votes
1 answer
105 views

Solve the integration by parts [closed]

Solve this integration by parts ∫ln(x+1)/x dx
carolusquintae's user avatar
1 vote
1 answer
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Solving a first order linear non-homogeneous ODE

can someone help me figure out how this differential equation \begin{equation} y' = p(t)(1-y) \tag{1} \end{equation} gives the following solution? \begin{equation} y = 1 + \exp\left[-\int_0^tp(s)ds\...
Jeffrey Alido's user avatar
2 votes
0 answers
74 views

Evaluation of $\displaystyle \int \frac{x^2}{(x^4-1)\sqrt{x^4+1}}dx$ [duplicate]

Evaluation of $\displaystyle \int \frac{x^2}{(x^4-1)\sqrt{x^4+1}}dx$ What I try : Using Substution, $\displaystyle x=\frac{1-t}{1+t}$ and $\displaystyle dx =\frac{2}{(1+t)^2}dt$ also using $\...
jacky's user avatar
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1 vote
2 answers
137 views

Integrate $\frac1{1 + ax^2}$ using partial fractions decomposition

I'm trying to solve the following integral: Integral of: $\int\frac{1}{1+ax^2}dx$ Where $a$ is some positive constant. We can't of course use basic U Substitution as the derivative of $1 + ax^2$ is $...
Aviv Cohn's user avatar
  • 449
2 votes
1 answer
86 views

How to solve $\int \frac{1}{\sqrt{x^2+x-1}}dx$? [duplicate]

How can one evaluate $\int \frac{1}{\sqrt{x^2+x-1}}dx$? Our professor has given us this integral to think about and I've no idea what to do to solve it. I understand there's probably a substitution ...
drafty nene's user avatar
1 vote
0 answers
75 views

Need help in evaluating this definite integral [duplicate]

I have been trying to solve this integral. Have not had any success. Any help would be appreciated. I know the answer is $2\pi^2$, I need the solution $$ \int\limits_0^\infty\frac{\ln^2\left(x\right)}{...
zynox's user avatar
  • 152
1 vote
2 answers
102 views

Integrating $\int\frac{\cos(\omega t)\gamma e^{-\gamma t}}{\omega}dt$

How to integrate $$\int\frac{\cos(\omega t)\gamma e^{-\gamma t}}{\omega}dt,$$ where $\gamma, \omega \neq 0$. I tried using substitution $u=\omega t$, $du=\omega dt$ and got $\frac{1}{\omega^2} \int \...
gujaral's user avatar
  • 146
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1 answer
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Sufficient condition for convergence of integral

Given a function $f\colon \mathbb R \to \mathbb R$ and that $$ \lim_{t\to \infty}\int_{-t}^tf(x) \ dx=1, $$ does the integral $$ \int_{-\infty}^{\infty}f(x) \ dx $$ necessarily converge? I've been ...
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-1 votes
1 answer
46 views

Does $\frac{\int^a_b f(x)dx}{\int^a_b g(x)dx}= \int^a_b \frac{f(x)}{g(x)}dx$ for $g(x)\neq 0$ with $x\in [a,b]$

Does $$\frac{\int^a_b f(x)dx}{\int^a_b g(x)dx}= \int^a_b \frac{f(x)}{g(x)}dx$$ is $$g(x)\neq 0$$ for $x\in [a,b]$ I came up with this question when I am learning Mellin's transform. I am not sure if ...
dumbdumb1234's user avatar
2 votes
0 answers
52 views

Inverse Mellin's tranform of Pochhammer(quotient of Gamma functions)

I am trying to evaluate the reverse Mellin's transform for the pochhammer symbol. I got to this equation: $\frac{1}{2\pi \cdot i} \int_{\sigma-i\infty}^{\sigma+i\infty}\frac{\Gamma(s)} {\Gamma(s+k+1)} ...
dumbdumb1234's user avatar
0 votes
1 answer
73 views

Evaluating the integral as a function of a matrix and a vector

Evaluate the integral $$ \int{e^{{-1 \over 2}x^T{Yx+z^Tx}}dx} $$ as w.r.t $Y$ (matrix) and $z$ (vector) My attempt I have never seen such a problem before, and I have very unsure how to start. Since I'...
PROTechThor's user avatar
1 vote
1 answer
60 views

integral of $\int \frac {dx}{2\sin(x)^2 + 3\cos(x)^2} $

The answer is supposedly $$\frac {1}{\sqrt{6}} \arctan\left(\sqrt{\frac {2}{3}}\tan x\right) + C$$ So I need to get it into form $$\int \frac{\mathrm dx}{a^2+x^2} $$ but I am not sure what identities ...
Kryptic Coconut's user avatar
5 votes
2 answers
123 views

How to evaluate $\int\left(\frac{\sin(x)}{2\sin(x)- x(1+\cos(x))}\right)^2dx$?

I saw this problem: $$\int\left(\frac{\sin(x)}{2\sin(x)- x(1+\cos(x))}\right)^2dx$$ I tried to solve this problem and I found a strange and unsatisfactory solution using differential equations. Let $$...
pie's user avatar
  • 4,314
-2 votes
1 answer
37 views

Integration of multivariate odd symmetric function [closed]

If $f(\mathbf{x}): R^p \rightarrow R $ is a multivariate odd-symmetric function in the sense that $f(\mathbf{x}) = -f(-\mathbf{x})$ for any $\mathbf{x}$ and it is absolutely integrable, does it ...
wutai's user avatar
  • 25
5 votes
1 answer
235 views

How to find the coefficient of $x^k$ in the expression $\prod_{p=1}^n (x^p+1)^p$?

This Question asked on math over flow I tried to find the indefinite integral $$ f_n(x)=\int \prod_{k=1}^n \cos^k(kx)dx$$ by using Euler's formula and put $x=\frac{\ln y}{2i}$ I got $$ f_n(x)=-i2^{-\...
Faoler's user avatar
  • 1,321
0 votes
0 answers
42 views

Properties of the Riemann integral - is my proof correct?

I've attempted to tackle the following problem and I'm not sure about my solution: $f:\:\left[-1,1\right]\:\rightarrow \:\mathbb{R}$ is an odd and integrable function. Show that: $$\int _{-1}^1\:\:...
Blabla's user avatar
  • 169
1 vote
2 answers
125 views

Integral of $\sqrt{\tan x}$

Evaluate this integral: $\int\sqrt{\tan x } \ dx$ So I had an idea, which just looks too simple to be true, but correct me please, I believe it has to be wrong... Rewrite as: $$\int\sqrt\frac{\sin x}{...
Petr Mášek's user avatar
0 votes
1 answer
25 views

How $\frac{1}{2a} \log {|\frac{x+a}{x-a}|} + C = \frac{1}{2a} \log {|\frac{a+x}{a-x}|} + C$

I know how $\int{\frac{dx}{x^2-a^2}} = \frac{1}{2a} \log {|\frac{x-a}{x+a}|} + C$ Using the above derivation, I derived $$\int{\frac{dx}{a^2-x^2}} = -\int{\frac{dx}{x^2-a^2}} = \frac{1}{2a} \log {|\...
Cinverse's user avatar
  • 181
3 votes
3 answers
324 views

u-substitution of indefinite integrals without algebraically manipulating differentials

Take for example $$\int 2x \cos(x^2)dx$$ Its easy to see that this is the result of chain rule. If we take $u=x^2$ then $dx = du/2x$ and then we get $$\int \cos(u)du$$ and it is simple from there. But ...
Kryptic Coconut's user avatar

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