Question

Open the pdf file "exp 2 measurement." Select the integrals below that are convergent.

a) ∫(0 to [infinity]) e^(-x) dx
b) ∫(1 to [infinity]) 1/x dx
c) ∫(-[infinity] to [infinity]) x^2 dx
d) ∫(0 to 1) 1/x dx

Answer 1

The convergent integrals are ∫(0 to ∞) e^(-x) dx and ∫(-∞ to ∞) x^2 dx.

Step-by-step explanation:

To determine which integrals are convergent, we need to evaluate each integral and see if the result is a finite number.

a) The integral ∫(0 to ∞) e^(-x) dx can be evaluated using integration by parts and will converge to 1.

b) The integral ∫(1 to ∞) 1/x dx can be evaluated as ln(x) from 1 to ∞ which diverges to infinity.

c) The integral ∫(-∞ to ∞) x^2 dx is a definite integral and the result will be a finite number.

d) The integral ∫(0 to 1) 1/x dx can be evaluated as ln(x) from 0 to 1 which diverges to negative infinity.

Therefore, the integrals that are convergent are a) and c).