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

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1
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1answer
46 views

Evaluate the integral of : $\int \:\frac{e^{2x}}{1+e^{3x}}dx$

Evaluate the integral of : $\int \:\frac{e^{2x}}{1+e^{3x}}dx$ So I start by making : $e^{2x}=t$ and this gives me the following : $\frac{1}{2}\int \:\frac{dt}{1+t\sqrt{t}}$ Then using symbolab ...
1
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1answer
31 views

Natural Logs and Anit-Derivatives are kicking me

I am given a problem involving rates of flow, $F(t)=\frac{t+7}{2+t}$ is the rate at which a bucket is being filled. The same bucket is being emptied at a rate given by $E(t)=\frac{\ln(t+4)}{t+2}$. My ...
0
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3answers
69 views

Why does $z\mapsto \exp(-z^2)$ have an antiderivative on $\mathbb C$?

Why does $z\mapsto \exp(-z^2)$ have an antiderivative on $\mathbb C$? So far I have seen the following results: If $f\colon U\to\mathbb C$ has an antiderivative $F$ on $U$ then ...
-1
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2answers
78 views

Easiest way to solve this integral [on hold]

I was solving this problem from a calculus textbook and I got stuck at this particular problem. I tried to put it into Integral Calculator after I was unable to solve it, but now I wonder if there is ...
2
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1answer
61 views

Evaluate $\int \sin(3x)\cos(4x) \,dx$

Evaluate $$\int \sin(3x) \cos(4x) \; \mathrm{d}x$$ I do not know how to solve this as a whole. I tried all the substitutions known to me.
0
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1answer
72 views

Simple yet challenging integral, can it be solved analytically, and if so, the answer.

I'm trying to find solutions to the 3 following integrals. The first 2 are of the same form, only varying by a constant in the numerator within the cosine, and yes, x is a constant in the first one. ...
3
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0answers
32 views

what is the integral of square root of sin(x),cos(x),sec(x),cosec(x)?

I have solved integral of square root of tan(x),cot(x).but i could not solve these questions.I know first we put tan(x)=t(any variable) and solve further.
0
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1answer
30 views

Indefinite INTEGRAL fraction [duplicate]

Compute the indefinite integral Irrational: $$\int \frac{x-1} {1 + \sqrt{x^2+2x-3}}dx$$ Help me pls. What do I need to do then? fraction isn't simplified enter image description here
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0answers
8 views

Which probability distribution(s) $f(x)$ allow for a closed form solution to $\int\left(x-a\right)^{-\gamma}f\left(x\right)dx$?

I'm trying to find if there is a specific probability distribution $f\left(x\right)$ (or many) such that the following integral $$\int\left(x-a\right)^{-\gamma}f\left(x\right)dx$$ has a closed form ...
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0answers
40 views

Indefinite INTEGRAL fraction [on hold]

Compute the indefinite integral Irrational: $$\int \frac{x-1} {1 + \sqrt{x^2+2x-3}}dx$$ multiplying by conjugate enter image description here Help me pls. What do I need to do then? fraction isn't ...
1
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2answers
26 views

most general antiderivative involving sec x

I'm stumped on how to get the most general antiderivative, $F(x)$, of $f(x)=e^x+3secx(tan x + sec x)$. First, I split the equation on addition, since $\int[f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx$ ...
3
votes
1answer
51 views

Antiderivative of $\arctan(-x^2)$

As I said in the title I'm trying to find an antiderivative of $$f(x)=\arctan(-x^2)$$ I am aware that e.g. WolframAlpha can find one, but I have no clue how to do it by hand. Can anyone give me a ...
7
votes
3answers
120 views

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

Evaluate $$\int \frac{1-x}{(1+x)\sqrt{x+x^2+x^3}}dx$$ i used substitution $x=\tan^2 y$ so $dx=2\tan y \sec^2 y dy$ so the integral becomes $$I=\int\frac{2\cos 2y\: \tan y\: \sec^2 y ...
1
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0answers
45 views

Hint for solving an indefinite integral

I would like to solve the following integral $$\int \frac{dx}{x^2 \sqrt[4]{(a-x^2)(b+x^2)}},$$ where $a$ and $b$ are the real constants. My attempt: $$\sqrt[4]{(a-x^2)(b+x^2)} = \sqrt[4]{-\bigg[x^4 ...
0
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0answers
21 views

symmetry of two IID random variables [duplicate]

Suppose that $X$ and $Y$ are independent and identically distributed. The claim is that $P(X<Y)=P(X>Y)=1/2$. How do I prove this? My attempt Since they are IID $f_X=f_Y$. So ...
4
votes
2answers
56 views

Difference in use between $d$, $\partial$, $\operatorname d$, $\varDelta$ and $D$ for derivatives.

While reading different sources on implicit differentiation (and thereafter differentiation in general), I came across many different "d's" being used for (or similar to) the familiar ...
0
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1answer
40 views

How to simplify this ln?

I am solving this problem $$\int {\frac{1}{\sqrt{9x^{2}-25}}dx}$$ the step I got stuck, $$\frac13 \ln{\left(\frac{3x+\sqrt{9x^2-25}}{3}\right)} $$ and in my textbook the answer was $\frac13 ...
1
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3answers
91 views

Integral of $\frac{x}{(1-x^3)\sqrt{1-x^2}}$

In the pool of difficult (at least to me) integrals I've been trying to solve this one: $$\int\frac{x}{(1-x^3)\sqrt{1-x^2}}dx$$ Since Wolfram Alpha has been helpful with all the other integrals ( at ...
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0answers
34 views
+50

Integration and differentiation of Fourier series

I am interested in the properties of Fourier series under integration and differentiation, and I've noticed a "strange" phenomenon. Suppose I have a Fourier series which I Integrate, and suppose that ...
1
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2answers
86 views

Evaluation of $\int \frac{\sqrt{\sin ^4x+\cos ^4x}}{\sin ^3x. \cos x }dx$

Evaluate the following integral: $$\int \frac{\sqrt{\sin ^4x+\cos ^4x}}{\sin ^3x. \cos x }dx$$ where $x \in \big(0,\frac{\pi}{2} \big)$ Could some give me hint as how to approach this question? I ...
3
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0answers
47 views

Integration by guessing the form of the numerator

I sometimes see integrands in textbooks with a square in the denominator, like this one: $$\int\frac{x^2}{\left(x\sin\left(x\right)+\cos\left(x\right)\right)^2} dx$$ Often, these integrands are ...
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4answers
127 views

Computation of $\int \tanh^5 2x \textrm d x$

Really struggling with a problem here: I need to find $\int \tanh^5 2x \textrm d x$ - absolutely no idea how to do it. I tried splitting into $\tanh^2 2x,\tanh^2 2x, \tanh 2x$, and tried using ...
0
votes
1answer
47 views

Antiderivative of $\frac{e^{x-1}}{x}$

I am trying to calculate $$\int e^{x-1} \frac{1}{x} dx $$ but got stuck. I first used substitution to rewrite the integral as $$\int e^{x} \frac{1}{x+1} dx .$$ Then I made the substitution $\sqrt{x} ...
1
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1answer
21 views

Justification of this step involving Gaussian Integral

I have came across the following step, I suspect it is true but I am not sure how it is justified. $$ I = \int_{-\infty}^{\infty} \frac{dk}{2\pi} \exp{\left(i k (x_0 - x_t) - \frac{k^2}{2} ...
1
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0answers
60 views

$\int\frac{{11}/{10}}{2x+2}dx$ what is wrong

$$\int\frac{\frac{11}{10}}{(2x+2)}dx$$ $$\frac{11}{10}\int \frac{dx}{(2x+2)}$$ $$t=2x+2$$ $$dt=2dx$$ $$dx=\frac{dt}{2}$$ $$\frac{11}{10}\int\frac{1}{t}\frac{dt}{2}$$ ...
0
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1answer
13 views

Example of primitivable function

Recently we learned the Leibniz-Newton formula, that linked the primitive with the integral. I know that Leibniz-Newton can be applied when the function is both integrable and primitivable, but not ...
0
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2answers
57 views

Integrate $\sin(3x)\cos(3x)$

Integrate $\sin(3x)\cos(3x)$ I looked at various answers on different sites but still do not understand how to use the u-substitution method in this question or the double angle rule.
4
votes
1answer
68 views

Integral of $\dfrac{1}{x\sqrt{x^2-1}}$

I am very confused by this. I know that the derivative of $\text{arcsec}(x)$ is $\dfrac{1}{|x|\sqrt{x^2-1}}$. However, if you plug in the integral of $\dfrac{1}{x\sqrt{x^2-1}}$ into wolfram alpha it ...
1
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3answers
138 views

How to integrate $x^2\sin(x^2)$?

$$\int x^2\sin(x^2)dx$$ Methods I know of: Reversing the chain, substitution, parts. Any help greatly appreciated. Edit: Note that the integral is not $x^2\ast(\sin x)^2$
3
votes
3answers
53 views

How to solve this integration that I got from differential linear equation?

I worked differential linear equation and at end of equation I got this integral.Can someone give me a hint to do this: $$\int \frac{1}{\left(u^2+1\right)^2}du$$
3
votes
1answer
120 views

Integration of $\int \frac{x^2+20}{(x \sin x+5 \cos x)^2}dx$

How do we integrate $$\int \frac{x^2+20}{(x \sin x+5 \cos x)^2}dx$$ Could someone give me some hint for this question?
1
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3answers
115 views

Solving the integral of $sin^n(x)$ [duplicate]

I need to find an expression for $\int\sin^n(x)dx$ with $n\in\mathbb{N}$. I am interested in answers that tell me how to start solving this problem, as much as I am interested in the actual solution. ...
3
votes
3answers
557 views

How to calculate this integral with square roots: $\int\frac{ \sqrt{x+1} }{ \sqrt{ x-1 }} \, dx$

How would you calculate this integral: $$\int_{}\frac{ \sqrt{x+1} }{ \sqrt{ x-1 }} \, dx$$
0
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1answer
35 views

What is the indefinite integral of $\sqrt[2n] {\tan \left( x\right) }$

I have already solved the following integrals $\sqrt {\tan \left( x\right) }$ and $\sqrt[4] {\tan \left( x\right) }$ (the last one with some help) so I want to know if it's possible to have a solution ...
6
votes
1answer
75 views

Evaluating an indefinite integral using complex analysis

Using tools from complex analysis, I have to prove that $$ \int_0^{\infty} \frac{\ln x}{(x^2 + 1)^2}\,dx = - \frac{\pi}{4}.$$ But I'm not really sure where I should start. Any help would be ...
3
votes
2answers
40 views

$f(x)=x^a$ Definite Integral

Consider $f(x)=x^a $ Now $\int_0^1 x^a = 1/(1+a)$ gives the area bounded by the function, $x $ axis, $x=0$ and $x=1$. Now consider $a<-1$ On LHS the function is positive for all $0<x<1$ ...
1
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3answers
79 views

What is the indefinite integral of $\sqrt[4] {\tan \left( x\right) }$ [closed]

$$\int\sqrt[4] {\tan \left( x\right) } dx$$ I'm really stuck right now with this integral, so any kind of advice would be appreciated. Perhaps a pretty nifty substitution that can save me from ...
0
votes
1answer
46 views

$\int cos^5(x) dx$ with reduction formula

I'm trying to calculate $$\int cos^5(x)dx$$ with the reduction formula. $$\int cos^5(x)dx=\frac{1}{5}cos^4(x)sin(x)+\frac{4}{5}\int cos^3(x)dx$$ then $$\int ...
2
votes
2answers
62 views

How would I compute this integral with a ceiling function?

It seems simple, but it is not, really. How would I calculate this ($\lceil a \rceil$ denotes the ceiling function)? $$\int{\lceil x+2 \rceil}\ln x\,\,dx$$ First, I noticed that $\lceil x+2 ...
0
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2answers
25 views

Solve simple indefinite integral using basic table

I am trying to find the indefinite integral of $\frac{1}{\sqrt{3x^2-1}}$ using the table of basic indefinite integrals (no substitution or integration by parts). I considered using the formula ...
2
votes
3answers
47 views

How we get $\frac{3}{2} \int e^u du$ from $3 \int e^{2x} dx$

I cannot understand how we get $\frac{3}{2} \int e^u du$ from $3 \int e^{2x} dx$ after substituting $u = 2x$ and $du = 2dx$. Wolfram Pro gives this substitution step never explaining how we get ...
0
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1answer
24 views

Differential equation with cosine squared

I'm having some trouble solving the differential equation $$\ \frac{dy}{dx}= cos^2(\frac{\pi y}{2})$$ when y = 0.5 its x=0 and I need to find the value of x when y=2.5, anyone able to help?
1
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1answer
32 views

Is the indefinite integral of a piecewise continuous function a continuous function?

I had looked around on the web and can't find much information related to the integration of piecewise continuous functions. Let's say we have a simple function $$f(x)= \begin{cases} 0 & ...
0
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2answers
50 views

Tips on how to integrate $\int x\sin{(1+x^2)}\sin{(1-x^3)}\ dx$

So, I'm completely stuck at integrating this integral: $$\int x\sin{(1+x^2)}\sin{(1-x^3)}\ dx$$ The solution should supposedly be: $-\frac{1}{4}x^2\cos{(2x^2)}+\frac{1}{8}\sin{(2x^2)}+C$. Using ...
3
votes
2answers
70 views

How to solve $\int\frac{e^{-2x}\sin^2x}{1-\sin{2x}}dx$

Consider the integral $$\int\frac{e^{-2x}\sin^2x}{1-\sin{2x}}dx$$ approch: $$\int\frac{e^{-2x}\sin^2x}{1-\sin{2x}}dx=\int\frac{e^{-2x}\sin^2x}{(\cos x-\sin x)^2}dx=\int \frac{e^{-2x}\sin^2x}{\cos ...
4
votes
2answers
160 views

How to solve $\int \dfrac{x^5\ln\left(\frac{x+1}{1-x}\right)}{\sqrt{1-x^2}} dx$

Consider the integral $$\int \dfrac{x^5\ln\left(\frac{x+1}{1-x}\right)}{\sqrt{1-x^2}}dx$$ How to start integrating? Any hint would be appreciated.
3
votes
1answer
285 views

Finding $\int\frac{\sqrt{1+\sqrt{1+\sqrt{1+\cos(2\sqrt{x+5})}}}}{\sqrt{x}} dx$

The following integral is well posed? we must correct? $$\int\frac{\sqrt{1+\sqrt{1+\sqrt{1+\cos(2\sqrt{x+5})}}}}{\sqrt{x}}dx$$ Any hint would be appreciated.
1
vote
1answer
54 views

Other way of integration of $(\sin^2 x)$

I was wondering if there is any other way of integrating $(\sin^2 x)$ without using formula of $\cos 2 x$. Please avoid expansion series in your answers. Thanking in advance.
1
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0answers
23 views

Generalization of anti-derivative

I am familiar with the integration of a function $f:D\subset\mathbb{R}^n\to \mathbb{R}$ with respect of some variable $x_i \,\,1\leq i \leq n $. For example: $$\int x\sin (y)\, ...
0
votes
0answers
41 views

Calculate non-elementary integrals

I'd like to calculate $\int_{-\infty}^{\infty}\sin(x^2)dx$ and $\int_{-\infty}^{\infty}\cos(x^2)dx$ I think it may be possible to do it by using the fact that: $$\int_{\gamma}^{} e^{-z^2}dz =0$$ For ...