hw 9 question 10

Question

Integrate (using substitution).

Answer 1

= the integral of 256t^7 - 192t^5 + 48t^3 -4t

= (256/8)t^8 - (192/6)t^6 + (48/4)t^4 - (4/2)t^2 + C where is an unknown constant

= 32t^8 - 32t^6 +12t^4- 2t^2 + C

= 2t^2(16t^6 -16t^4 +6t^2 -1) + C

check & take the derivative

= 256t^7 - 192t^5 + 48t^3 -4t

=4t(64t^6 - 48t^4 +12t^2 -1)

=4t(4t^2-1)(16^4-8t^2+1)

=4t(4t^2 -1)(4t^2 -1)^2

=4t(4t^2 -1)^3

or let u=(4t^2 - 1) du=8tdt,

integral of 4t(4t^2 -1)^3 = u^3du/2 = (1/2)integral of u^3du = (1/2)u^4/4 = (1/8)u^4 =

(1/8)(4t^2 -1)^4 + C where C= an unknown constant

Answer 2

Solution:

let u = (4t² -1), du = 8t dt, dt = 1/8 du substitute in integral

1/8 ∫ 4 u³ du = 1/2 ∫ u³ du = 1/2 ( u³⁺¹/ 3+1) + C = 1/2( u⁴/4) + C = 1/8 u⁴ + C

resubstitute 4t² -1 into u, = 1/8( 4t²-1)⁴ + C

to check find the derivative of the integral wrt to t = 1/8(4)( 4t²-1)³ ( 8t) = 4t( 4t²-1)³ same as in the integrand