What is the definite integral of X to the nth power from 0 to 1

What is the definite integral of X to the nth power from 0 to 1

The derivative of 1 / (n + 1) * x ^ (n + 1) is x ^ n. in fact, to find the definite integral of the nth power of X on (0,1) is to find the area of the region

For (E's negative s times x power) times X's n power, find the definite integral from 0 to positive infinity~

∫[0,+∞)x^n*e^(-sx)*dx
=1 / S ^ (n + 1) ∫ [0, + ∞) T ^ [(n + 1) - 1] * e ^ (- t) DT (let t = SX)
=1/s^(n+1)*Γ(n+1)
=n!/s^(n+1)

The definite integral of the x power of E from negative infinity to zero
How to ask?

It's 1
The original function is e ^ x = 0, e ^ 0 = 1
x=-inf e^-inf=0
So it's 1

How to find the generalized integral of the negative x square power of E from negative infinity to positive infinity
That is to say, the upper and lower limits of the integral are negative infinity and positive infinity respectively, and the integrand is e ^ - x square. How to find the integral (the answer is the root sign PAI)

I=[∫e^(-x^2)dx]*[∫e^(-y^2)dy]
=∫∫e^(-x^2-y^2)dxdy
Convert to polar coordinates
=[∫ (0-2 π) Da] [∫ (0 - + infinity) e ^ (- P ^ 2) PDP]
=2 π * [(- 1 / 2) e ^ (- P ^ 2) | (0 - + infinity)]
=2π*1/2
=π
∫e^(-x^2)dx=I^(1/2)=√π