1
vote
1answer
45 views

$\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx<\infty $ for some large $n$?

Fix $y\in \mathbb R.$ Define, $$I(y)=\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx.$$ My Question is: Can we show that $I(y)<\infty$ for some large $n\in \mathbb N$ ? If yes, what is a value ...
5
votes
1answer
93 views

Determining the best possible constant $k$, for an Integral Inequality

If $f : [0,\infty) \to [0,\infty)$ is an integrable function, then what is the best possible constant $k$, for which the following ineqality holds: $$\int_0^{\infty}f(x)dx \leq ...
7
votes
5answers
386 views

Solving a separable differential equation

Solve the differential equation: $$y'=\frac{1-y^2}{1-x^2}$$ My book says the solution is: $$y=\frac{x+c}{cx+1},$$ where $c$ is a constant. It's been ten minutes I tried to verify if it was correct ...
6
votes
3answers
153 views

How to find integral $\underbrace{\int\sqrt{2+\sqrt{2+\sqrt{2+\cdots+\sqrt{2+x}}}}}_{n}dx,x>-2$

Find the integral $$\int\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots+\sqrt{2+x}}}}}_{n}dx,x>-2$$ where $n$ define the number of the square I know this if $0 \le x\le 2$, then let ...
5
votes
2answers
166 views

How to integrate $\int \frac{1}{\sin^4x + \cos^4 x} \,dx$?

How to integrate $$\int \frac{1}{\sin^4x + \cos^4 x} \,dx$$ I tried the following approach: $$\int \frac{1}{\sin^4x + \cos^4 x} \,dx = \int \frac{1}{\sin^4x + (1-\sin^2x)^2} \,dx = \int ...
4
votes
4answers
117 views

How to integrate $\int \frac{\cos x}{\sqrt{\sin2x}} \,dx$?

How to integrate $$\int \frac{\cos x}{\sqrt{\sin2x}} \,dx$$ ? I have: $$\int \frac{\cos x}{\sqrt{\sin2x}} \,dx = \int \frac{\cos x}{\sqrt{2\sin x\cos x}} \,dx = \frac{1}{\sqrt2}\int \frac{\cos ...
5
votes
2answers
189 views

How to solve $\int \frac{\,dx}{(x^3 + x + 1)^3}$?

How to solve $$\int \frac{\,dx}{(x^3 + x + 1)^3}$$ ? Wolfram Alpha gives me something I am not familiar with. I thought that the idea was using partial fractions because $x^3$ and $x$ are ...
3
votes
3answers
113 views

How to solve $\int \frac{x^4 + 1 }{x^6 + 1}$?

How to solve $\int \frac{x^4 + 1 }{x^6 + 1}$ ? The numerator is a irreducible polynomial so I can't use partial fractions. I tried the substitutions $t = x^2, t=x^4$ and for the formula $\int ...
0
votes
1answer
80 views

How to integrate $\int\frac{xe^{tan^{-1}x}}{(1+x^2)^\frac{3}{2}}\,dx$? [duplicate]

Original question: (updated question in section EDIT) How to evaluate the following integral? $$\int\frac{1}{(1+x^2)^\frac{3}{2}}\,dx$$ I tried substitution: $$ t = 1+x^2 = \varphi \\ ...
1
vote
2answers
55 views

a question about integral with parameter variables?

I have a problem proving $$\int_{0}^\infty dx {\left(\int_{0}^\infty e^{-x^2t}\sin t\, dt\right)}=\int_{0}^\infty dt\left( \int_{0}^\infty e^{-x^2t}\sin t\, dx\right)$$. I have been struggling for ...
0
votes
2answers
87 views

Integral $\int\sqrt{\sin2x}\operatorname d\!x$

I tried all substitutions but failed. I need assistance to evaluate that indefinite integral. $\int\sqrt{\sin2x}\operatorname d\!x$
0
votes
1answer
29 views

Find Indefinite of root function

I don't know how to find this strange integral $\int{\sqrt{\dfrac{x-4}{x+2}}\dfrac{dx}{x+2}}$ Please help me solve this problem
1
vote
3answers
91 views

If $f:[0,1]\rightarrow\mathbb{R}$ is a function such that $f=0$ over a dense set in $[0,1]$ so $\int_0^1 f=0$?

If $f:[0,1]\rightarrow\mathbb{R}$ is a Riemann-Integrable function such that $f=0$ over a dense set in $[0,1]$ so $\int_0^1 f=0$? I'm thinking about it but without progress. Would someone ...
1
vote
2answers
282 views

The unbeatable $\int e^{1/\cos(x)} dx$ integral

Is there any way to express this in non-elementary functions? $$ \int e^{1/\cos(x)} dx$$ And/or to calculate this definite integrals? $$ \int_{-\pi/2}^{\pi/2} e^{1/\cos(x)} dx$$ $$ ...
2
votes
3answers
114 views

Find the indefinite integral $\int\frac{(x+1)e^x}{x(1+xe^x)}dx$

Find the indefinite integral $$\int\frac{(x+1)e^x}{x(1+xe^x)}dx$$ I feel like this function does not have an anti-derivative in the form of elementary functions.
7
votes
2answers
238 views

When we can change $\int$ and $\sum$ for indefinite integral?

I know, for example, that if the series $\displaystyle\sum_{n=1}^{\infty}f_n(x)$ consisting of integrable functions on a closed interval $[a, b] \subset \mathbb{R}$ converges uniformly on that closed ...
1
vote
1answer
168 views

Let $f:[0,1]→\mathbb{R} $with $f′(x) $continuous. It is known that $\int_{0}^{1} f(x)dx=0$.

Let $f:[0,1]→\mathbb{R}$ with $f'(x)$ continuous. It is known that $∫_0^1 f(x) dx=0$. Prove that $∀α∈[0,1]$, $$|\int_{0}^{\alpha} f(x) dx |≤ \frac{1}{8} sup_{(0≤x≤1)}|f'(x) |$$ My answer so far ...
9
votes
2answers
334 views

Evaluate $\int\sin(\sin x)~dx$

I was skimming the virtual pages here and noticed a limit that made me wonder the following question: is there any nice way to evaluate the indefinite integral below? $$\int\sin(\sin x)~dx$$ Perhaps ...
0
votes
1answer
165 views

Derivative of the indefinite integral and Lebesgue point

Give an example where the derivative of the indefinite integral exists at point that are not Lebesgue points.
5
votes
2answers
204 views

Evaluating $\int \cos(x) \sqrt{\sin(2 x)} dx$

Evaluate the following indefinite integral: $$\int \cos(x) \sqrt{\sin(2 x)} dx$$ Only hint I have is from W|A that expresses the integral in terms of a hypergeometric function and it looks ...
5
votes
2answers
302 views

Evaluating: $\int \frac{t}{\cos{t}} dt$

How would you evaluate the following indefinite integral? In fact, I did evaluate $\int \frac{\cos{t}}{t} dt$ by parametric integration and then I thought of this variant. $$\int \frac{t}{\cos{t}} ...