Integration

������������������������������������������������������������������ ������������������������������������ – ������������������������ ������������������������ ������������������������������������������������������ ������������������ ������������������������������������������������ ������������������������������������������������������ ������

TUTORIAL WORKSHEET 9 – INTEGRATION

1 . Find 1. =

2x 3 + 4e−x +

3 3 − dx x2 x

2x 3 + 4e−x +

3 3 − dx x2 x

1 4 3 x − 4e−x − − 3 ln |x| + C 2 x

2. Find 2. =

5

7x 3 − 3x −5 − e2x + π dx

5

7x 3 − 3x −5 − e2x + π dx 21 8 3 −4 1 2x x 3 + x − e + πx + C 8 4 2

3. Given that marginal revenue t weeks after a product is introduced is ,

dR −2000 = , dt t+1 2

if after 9 weeks revenue is 500, derive an epression for total revenue, R t . Given dR −2000 = dt t+1 2

To find R t Let u = t + 1 −2000 dt = t+1 2 ⟹ du =1 dt ⟹ du = dt u−2 du

−2000 du = −2000 u2

=

2000 +C u

Substituting, we get = 2000 +C t+1

Since R(t) = 500 when t = 9 2000 + C = 500 9+1 C = 500 − 200 = 300 Hence, R(t) = 2000 + 300 t+1

4. If Investment is the rate of change in the capital stock K over time t, given the net rate of investement is I t = 12t 3 and that K 0 is 25, what is the time path of capital stock K ? 4. Given I t = 12t 3 and K 0 is 25.

1 dK = 12t 3 dt 1 1

12t 3 dt = 9t 3 + C Since K(0) = 25 K t = 9t 3 + 25 is the time path of capital stock K

4 4

1

5. The marginal propensity to consume is

dC , where C is consumption and Y income. Given that dY

dC 0.1 = 0.8 + and that C = Y when Y = 100. Derive an expression for the consumption dY Y function C(Y).

5. Given 0.8 +

dC 0.1 = 0.8 + and that C = Y when Y = 100. dY Y 0.1 Y dY (Note I used K here as the constant of integrtation)

C Y = 0.8Y + 0.2 Y + K Since C = Y when Y = 100 100 = 0.8 100 + 0.2 100 + K 100 = 80 + 2 + K K = 18

Hence C Y = 0.8Y + 0.2 Y + 18

6. a. Find a. To find 5x − 7

8

dx

8

Hint: Integrating by Substitution...