Integration

Question 1

Integration introduction

Answer 1

The process of anti-differentiation can also be called integration. The result of integration is called the integral or anti-derivative.
* The anti-derivative of 𝑓 is denoted by ∫ 𝑓(𝑥) 𝑑𝑥
* The symbol ∫ is called an integral sign.
* 𝑓(𝑥) is the integrand.
* The 𝑑𝑥 specifies that this is the integral of 𝑓(𝑥) with respect to x

Question 2

Power rule

Answer 2

∫ 𝑎𝑥ⁿ 𝑑𝑥 =
[𝑎𝑥ⁿ⁺¹] / [𝑛+1] + 𝑐 , 𝑛 ≠ −1

Question 3

Properties of indefinite integral

Answer 3

∫[𝑓(𝑥) + 𝑔(𝑥)] 𝑑𝑥 = ∫ 𝑓(𝑥) 𝑑𝑥 + ∫ 𝑔(𝑥) 𝑑𝑥
∫[𝑓(𝑥) − 𝑔(𝑥)] 𝑑𝑥 = ∫ 𝑓(𝑥) 𝑑𝑥 − ∫ 𝑔(𝑥) 𝑑𝑥
∫ 𝑘𝑓(𝑥) 𝑑𝑥 = 𝑘 ∫ 𝑓(𝑥) 𝑑𝑥 𝑘 = constant

Question 4

The definition of a definite integral

Answer 4

For any function 𝐹 such that 𝐹′(𝑥) = 𝑓(𝑥)
For any function 𝐹 such that 𝐹′(𝑥) = 𝑓(𝑥)
∫ᵇₐ 𝑓(𝑥) 𝑑𝑥 = [𝐹(𝑥)]ᵇₐ = 𝐹(𝑏) − 𝐹(𝑎)

Question 5

Integral properties

Answer 5

∫ᵇₐ 𝑓(𝑥) 𝑑𝑥 = – ∫ᵃᵦ 𝑓(𝑥) 𝑑𝑥
∫ᵃₐ 𝑓(𝑥) 𝑑𝑥 = 0

Question 6

Integration of exponential functions 𝑒ˣ

Answer 6

∫eˣ dx = eˣ + c
∫eᵏˣ dx = 1/k eᵏˣ + c , k ≠ 0

Question 7

Integration of reciprocal 1/𝑎𝑥+b

Answer 7

If 𝑓(𝑥) = 𝑙𝑛(𝑥) then 𝑓′(𝑥) =1/𝑥
because integration is the same process as anti-differentiation it
follows that
∫1/𝑥 𝑑𝑥 = 𝑙𝑛|𝑥| + 𝑐 , 𝑥 ≠ 0

Note: The absolute (positive) value of is used because the function 𝑓(𝑥) = 𝑙𝑛(𝑥) is
only defined for positive values of .

Question 8

Integration of trigonometric functions

Answer 8

The integration of trigonometric functions can be done by inspection.
The opposite of trigonometric differentiation is the trigonometric integration

Question 9

Special cases for trig integration

Answer 9

∫sin²(x) dx = x/2 – 1/4 sin(2x) + c
∫cos²(x) dx = x/2 + 1/4 sin(2x) + c

Question 10

The trapezium rule

Answer 10

A better approximation for the area under a curve would be obtained if a straight line joining the coordinates across the top of each strip was used, thus approximating each strip to a
trapezium.

Area of Trapezium = ([𝑎+𝑏]/2) × ℎ

The trapezium rule with 𝑛 intervals states that
∫ᵇₐ 𝑓(𝑥) 𝑑𝑥 ≈ h/2 (y₀+2y₁+2y₂+ … +2yₙ₋₁+yₙ)

where h = b-a/n

Note: if the approximation was divided into more trapeziums the approximation would become more accurate

Question 11

Integration of 1 / x² + a²

Answer 11

y = tan⁻¹(x)
y’ = 1 / x² + 1

∫[1/ x² + 1] dx = tan⁻¹(x) + c
∫[1/ x² + a²] dx = 1/a tan⁻¹(x/a) + c

Question 12

Integrating quotient t 𝑘𝑓′(𝑥)/𝑓(𝑥)

Answer 12

∫f’(x)/f(x) dx = ln|f(x)| + c
∫kf’(x)/f(x) dx = k ln|f(x)| + c

Question 13

Integrating chain rule

Answer 13

∫ g’(x) x f’(g(x) = f[g(x) + c

Question 14

Integration by substitution

Answer 14

Steps:
1. Choose correct substitution
2. Change everything in terms of new variable and calculate new integral
3. Change back to original variable

Question 15

Integration of rational functions

Answer 15

When finding integrals such as
ax + b / cx + d and other expressions in fractional form it is sometimes best to divide using algebra long division first

Question 16

Integration with partial fractions

Answer 16

When given an equation that can be turned into a partial fraction, convert it and then integrate using standard integration procedures

Question 17

Integration by parts

Answer 17

Recall that if u and v are differentiable functions of x, uv = uv’ + v’u

∫uv’ = uv - ∫vu’

Note: the goal of this procedure is to end up with a single term or terms that can be integrated after the ∫