Linked Questions

7
votes
4answers
575 views

How to evaluate $\int_0^\infty \frac{1}{x^n+1} dx$ [duplicate]

Noticed that the integral $$\int_0^\infty \frac{1}{x^n+1} dx$$ is often approached with partial fraction decomposition, but this gets increasingly ugly as $n$ gets bigger. Is there a neat trick to do ...
6
votes
3answers
381 views

$\int_{0}^{\infty} \frac{1}{1 + x^r}\:dx = \frac{1}{r}\Gamma\left( \frac{r - 1}{r}\right)\Gamma\left( \frac{1}{r}\right)$ [duplicate]

As part of a recent question I posted, I decided to try and generalise for a power of $2$ to any $r \in \mathbb{R}$. As part of the method I took, I had to solve the following integral: \begin{...
2
votes
1answer
546 views

How to prove this integral problem: $\int_0^{\infty}\frac{dx}{1+x^n}=\frac{\pi}{n}\csc\frac \pi n$? [duplicate]

I dont know if this has already been asked. How to prove this integral $$\int_0^{\infty}\frac{dx}{1+x^n}=\frac{\pi}{n}\csc\frac \pi n\ {?}$$ $n\ge 2$ is a positive integer Frankly speaking i have ...
1
vote
1answer
119 views

Why is $\int_{0}^{\infty} \frac{1}{1 + x^t} \mathop{\mathrm{d}x}=\frac{\pi/t}{\sin(\pi/t)}$? [duplicate]

I don't know how to show $$ \int_{0}^{\infty} \frac{1}{1 + x^t} \mathop{\mathrm{d}x}=\frac{\pi/t}{\sin(\pi/t)} $$ for $t > 1$. I guess I'm supposed to use properties of gamma function and Euler's ...
0
votes
2answers
63 views

Integrate $\frac{dx}{x^{2n}+1}$ for $n \in \mathbb{N}$ from $-\infty$ to $\infty$ [duplicate]

How can one calculate $$ \int_{- \infty}^{\infty} \frac{dx}{x^{2n} + 1}, \;n \in \mathbb{N} $$ without using complex plane and Residue theorem?
0
votes
1answer
79 views

How do we prove such integral? [duplicate]

$$\int_0^\infty\frac 1{1+x^n}dx=\frac\pi n\csc\left(\frac\pi n\right)$$ If we want to prove the left side equal to the right side in this case, how do we start? How do we prove this definite integral?...
0
votes
1answer
48 views

How to calculate $I=\int_0^{\infty} \frac{dx}{x^4+a^4}$ [duplicate]

I understand it is an even function, which indicates $I=\frac{1}{2}\int_{-\infty}^{\infty} \frac{dx}{x^4+a^4}$ What should I do in the next step?
40
votes
7answers
3k views

Evaluate $\int_0^\infty\frac{\ln x}{1+x^2}dx$

Evaluate $$\int_0^\infty\frac{\ln x}{1+x^2}\ dx$$ I don't know where to start with this so either the full evaluation or any hints or pushes in the right direction would be appreciated. Thanks.
35
votes
8answers
3k views

Simpler way to compute a definite integral without resorting to partial fractions?

I found the method of partial fractions very laborious to solve this definite integral : $$\int_0^\infty \frac{\sqrt[3]{x}}{1 + x^2}\,dx$$ Is there a simpler way to do this ?
31
votes
10answers
6k views

Integral $I:=\int_0^1 \frac{\log^2 x}{x^2-x+1}\mathrm dx=\frac{10\pi^3}{81 \sqrt 3}$

Hi how can we prove this integral below? $$ I:=\int_0^1 \frac{\log^2 x}{x^2-x+1}\mathrm dx=\frac{10\pi^3}{81 \sqrt 3} $$ I tried to use $$ I=\int_0^1 \frac{\log^2x}{1-x(1-x)}\mathrm dx $$ and now ...
5
votes
7answers
300 views

Prove that $\int_0^{+\infty} \frac{\ln x}{a^2+x^2} dx = \frac{\pi\ln a}{2a}$

Is that true $$\int_0^{+\infty} \cfrac{\ln x}{a^2+x^2} dx = \cfrac{\pi\ln a}{2a},$$ where $a>0$ ? And how to compute it?
10
votes
6answers
271 views

Evaluation of $\lim_{n \to \infty} \int_{0}^\infty \frac{1}{1+x^n} dx$

$$L=\lim_{n \to \infty} \int_{0}^\infty \frac{1}{1+x^n} dx$$ $$\phi(x)=\lim_{n \to \infty}\frac{1}{1+x^n}$$ So I was just playing around with $\int_{0}^\infty \frac{1}{1+x^2} dx $ and it equals to $\...
7
votes
7answers
367 views

Find $\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$

How can we find the integral: $$\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$$ I tried to find and got it to be $\cfrac{\pi}{\sqrt2}$. Am I correct? Please help me with an ...
4
votes
3answers
314 views

How to evaluate $\int_{0}^{\infty}\frac{(x^2-1)\ln{x}}{1+x^4}dx$?

How to evaluate the following integral $$I=\int_{0}^{\infty}\dfrac{(x^2-1)\ln{x}}{1+x^4}dx=\dfrac{\pi^2}{4\sqrt{2}}$$ without using residue or complex analysis methods?
8
votes
2answers
535 views

Evaluate the integral $\int_{0}^{+\infty}\frac{dx}{1 + x^{1000}} $

Evaluate the integral \begin{equation} \int\limits_{0}^{+\infty}\frac{dx}{1 + x^{1000}} \end{equation} I tried using the change of variable, integration by parts, even wolframalpha... Nothing ...

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