evalualting definite integrals?

Question

∫ ^2 to 1 [x^2(x^3+3)^3]

okay the first step is to find u and du and they are

u=x^3+3
du=3x^2

then find the new x numbers in the integrand so...
(1)^3+3=4
(2)^3+3=11

∫ ^11 to 4 [u^3(1/3du)]

so now am confused what do I do next

Answer 1

take out the constant, 1/3, and anti-differntiate the intergrand then use the FTC to evaluate from 11 to 4.

Answer 2

I = ∫ x² (x³ + 3) ³ dx
let u = x³ + 3
du = 3x² dx
du / 3 = x² dx
I = (1/3) ∫ u³ du
I = (1/3) u^4 / 4
I = (1/12) (x³ + 3)^4 between limits of 4 and 11
I = (1/12) [ (11³ + 3)^4 - (4³ + 3)^4 ]
I = (1/12) [ 3.17 x10^12 - 2 x 10^7 ]
This is approx:-
I = (1/12) (3.17 x 10^12)
Comment
I feel that finding the integral is the important part of this question. The numerical part is basically using a calculator.

Answer 3

What you have so far seems fine

Since 1/3 is just a constant you can factor it out so you get that


∫ ^11 to 4 [u^3(1/3du)]= 1/3 ∫ ^11 to 4 u^3du

= 1/3 * 1/4 u^4 | _11 ^4 (this is my notation for from 11 to 4)

= 1/3 * 1/4 (4^4 - 11^4) and this is just a calculation of course.

I hope this helps

Answer 4

(1/3)∫^11 to 4 [u^3]
-(1/3)[ (u^4)/4 ]^11 to 4
-(1/3)[ (11^4)/4 - (4^4)/4 ]

Answer 5

Integrate the function with u as new variable; you have already done the hard stuff.