Time for some general integrals:

Calculus Level 3

∫ 0 ∞ x k 1 + x n d x = ? \large \int\limits_{0}^{\infty} \frac{x^k}{1+x^n} \, dx = \, ?

for: k , n ∈ Z k,n \in \mathbb{Z} and 0 ≤ k ≤ n − 2 0 \le k \le n-2

EDIT: The The π \pi s of the suggested answers in the denominator are supposed to be inside the s i n ( ∗ ) sin( * )

π n sin ⁡ ( k + 1 n ) π \frac{\sqrt{\pi}}{n\sin(\frac{k+1}{n}) \pi} π n sin ⁡ ( k + 1 n ) π \frac{\pi}{n\sin(\frac{k+1}{n}) \pi} k n + 1 \frac{k}{n+1} π ( 2 k + 1 ) sin ⁡ ( n 2 k + 1 ) π \frac{\pi}{(2 k+1) \sin(\frac{n}{2 k+1}) \pi} n π sin ⁡ ( k + 1 n ) π \frac{n \pi}{\sin(\frac{k+1}{n}) \pi} n 2 sin ⁡ ( k + 1 n ) \frac{n}{2 \sin(\frac{k+1}{n})}

Alisa Meier by Alisa Meier , 24, Germany

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1 solution

Substitute x = t 1 n \displaystyle x=t^{\frac{1}{n}} to get 1 n B ( k + 1 n , 1 − k + 1 n ) = π n sin ⁡ ( π ( k + 1 ) n ) \displaystyle \frac{1}{n}\Beta\left(\frac{k+1}{n},1-\frac{k+1}{n}\right) = \frac{\pi}{n\sin\left(\frac{\pi(k+1)}{n}\right)}

Here B ( x , y ) \Beta(x,y) denotes the Beta Function .

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