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### 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 ...
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{...
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 ...
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### 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 ...
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?
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### 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?...
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### 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?
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### 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.
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### 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 ?
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### 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 ...
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?
$$L=\lim_{n \to \infty} \int_{0}^\infty \frac{1}{1+x^n} dx$$ $$\phi(x)=\lim_{n \to \infty}\frac{1}{1+x^n}$$ So I was just playing around with $\int_{0}^\infty \frac{1}{1+x^2} dx$ and it equals to $\... 7answers 367 views ### Find$\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 ... 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? 2answers 535 views ### Evaluate the integral$\int_{0}^{+\infty}\frac{dx}{1 + x^{1000}} \$
Evaluate the integral $$\int\limits_{0}^{+\infty}\frac{dx}{1 + x^{1000}}$$ I tried using the change of variable, integration by parts, even wolframalpha... Nothing ...