If n is a non-negative integer?

Question

If n is a non-negative integer, then ∫ x^n dx from 0 to 1=∫ (1-x)^n dx from 0 to 1 for?

 

a) no n

b) n even, only

c) n odd, only

d) nonzero n, only

e) all n

 

Answer: E

 

 

Answer 1

Use u-substitution: u = 1-x => x = 1-u, and dx = -du

∫ x^n dx from 0 to 1

= ∫ (1-u)^n -du from 1 to 0

= ∫ (1-u)^n du from 0 to 1, switching the lower limit and the upper limit

= ∫ (1-x)^n dx from 0 to 1, substitute x = u

 

Therefore, it is true for all n.

 

Answer: e)