Integration

Question 1

|ax^n dx

Answer 1

|ax^n + 1/n+1 + c

Question 2

Definite Integrals

e.g |3 x^2 + x - 1 dx
-2

Answer 2

[x^3/3 + x^2/2 - x +c]
= ((3)^3/3 + (3)^2/2 - (3) + c)
- ((-2)^3/3 + (-2)^2/2 - (-2) + c)
= (27/3 + 9/2 - 3 + c) - (-8/3 + 4/2 + 2 + c)
= 12/6 +27/6 + 16/6
= 55/6

Question 3

Integrating sine and cosine

sinax

Answer 3

-1/a cosax + c

Question 4

Integrating sine and cosine

cosax

Answer 4

1/a sinax + c

Question 5

Area under a curve

|b F(b) - F(a) dx
a

e.g |3 x^2 - 2x + 2 dx
2

Answer 5

[x^3/3 - 2x^2/2 + 2x]
= ((3)^3/3 - 2(3)^2/2 + 2(3) +c)
- ((2)^3/3 - 2(2)^2/2 + 2(2) + c)
= (27/3 - 9 +6 +c) - (8/3 - 4 +4 +c)
= (9 - 9 + 6) - (8/3 - 4 +4)
= 10/3 units^2

Question 6

Integrating composite functions - chain rule

|(ax + b)^n dx

Answer 6

(ax + b)^n + 1/ n +1 x a

a = differentiated middle

Question 7

Integrating composite functions - chain rule

e.g

|(3x - 2)^5 dx

Answer 7

= (3x - 2)^6/ 6x3 + c
= (3x - 2)^6/18 + c

Question 8

Differential equations

e.g If dy/dx = 3x^2 - 6x passes through (2,-16)
Express y in terms of x

Answer 8

|dy/dx = |3x^2 - 6x
y = 3x^3/3 - 6x^2/2 + c

when x = 2, y = -16
-16 = (2)^3 - 3(2)^2 + c
-16 = 8 - 12 + c
-16 = -4 + c
c = -12

therefore, y = x^3 - 3x^2 - 12

Question 9

Area between two curves

e.g
upper - y = 6 - x - x^2
lower - y = x + 3

Answer 9

step 1 - calculate a & b by equaing the two equations

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

step 2 - set up the integral

|1 6 - x - x^2 - (x +3) dx
-3

step 3 - tidy up and intergrate

|1 3 - 2x - x^2 dx
-3
= [3x - 2x^2/2 - x^3/3 + c]
= (3(1) - 2(1)^2/2 - (1)^3/3 + c) -
(3(-3) - 2(-3)^2/2 - (-3)^3/3 + c)
= (3 - 1 - 1/3) - (-9 -9 + 9)
= 5/3 - (-9)
= 32/3 units^2