Please Evaluate?

Question

Evaluate
(from 1 to 0) 6x^5(x^6 + 25)^1/2 dx

Please show step by step work

a) [52(26)^1/2 + 250]/ 3 ~171.72

b) [52(26)^1/2 - 250]/18 ~0.84

c) 2(26)^1/2 -10 ~0.2

d) [52(26)^1/2 -250]/3 ~5.05

Answer 1

first, let

y= x^6 + 25 , so

dy= 6x^5 dx

then rearrange the terms in the integral:

[(x^6 + 25)^1/2] [(6x^5)dx]

from the regrouping it should be obvious that with the substitutions, we get:

y^1/2 dy

then we evaluate:

int. y^1/2 dy = 2/3 (y^3/2)

change y back to x:

y= x^6 + 25

int. 6x^5(x^6 + 25)^1/2 dx = 2/3 (x^6 + 25)^3/2 (from 0 to 1)

substitute your limits:

= 2/3 (x^6 + 25)^3/2 (x=1) - 2/3 (x^6 + 25)^3/2 (x=0)

= 2/3 (1^6 + 25)^3/2 - 2/3 (0^6 + 25)^3/2

= 2/3 (1 + 25)^3/2 - 2/3 (25)^3/2

= 2/3 [(26)^3/2 - (25)^3/2]

= 2/3 [26 * (26)^1/2 - 25 * (25)^1/2]

= 2/3 [26 * (26)^1/2 - 25 * 5]

= [52 * (26)^1/2 - 25 * 10] / 3

You should be able to check the answer from here. (d)