The FTCs used in evaluating integral of (2x-3)(x+1)dx from 0 to 1 are the following EXCEPT:

Question

3. The FTCs used in evaluating integral of (2x-3)(x+1)dx from 0 to 1 are the following EXCEPT:

a. Power rule

b. Constant multiple rule

c. Product rule

d. Constant rule

Answer 1

c. The product rule, since (1) after you foil out the product you have to use all of the other 3 rules in order to evaluate the integral, and (2) the product rule is not a rule for integrals, but rather instead a rule for derivatives, since the integral counterpart of the product rule for derivatives is called integration by parts.

Answer 2

S(2x-3)(x+1)dx

= S(2x^2 -x -3)dx (in some sense a "product rule" of FOIL, but not an integration rule)

= S2x^2dx - Sxdx -S3dx

= 2Sx^2dx -Sxdx (b: multiplation by constant rule) -3x +C (d: constant rule)

=2x^3/3 -x^2/2 (a: power rule) - 3x+C

= (2/3)x^3 - (1/2)x^2- 3x +C

evaluated from 0 to 1

= (2/3)(1^3) - (1/2)(1^2) -3(1) +C - [(2/3)(0)^3 -(0)^2 -3(0) + C] (difference rule)

= 2/3 -1/2 -3 +C - 0 +0+0-C

=2/3 -7/2

= 4/6 -21/6

= -17/6

=-2 5/6

all were used except C, the product rule, for derivatives, unless you consider FOIL a product rule, but not as an integration rule

FTC = Fundamental Theorem of Calculus