Chapter 16: Discovering Integrals

Question 1

State the rules for the integration of basic function types

Answer 1

1) ∫k dx = kx +c
2) ∫(xⁿ) dx = xⁿ⁺¹/n+1
3) ∫(eˣ) dx = eˣ+c
4) ∫(1/x) dx = ln|x|+c
5) ∫(cosx) dx = sinx + c
5) ∫(sinx) dx = -cosx+c

Question 2

Find the exact equation of a function
f(x) if given the gradient function and a particular point on the function

Answer 2

• Integrate f’(x) to find f(x) with the arbitrary constant c
• Sub in given point to find c
• Find exact equation f(x)

Question 3

State the rules for the integration of f(ax+b)

Answer 3

1) ∫eᵃˣ⁺ᵇ = 1/a (eᵃˣ⁺ᵇ) + c
2) ∫(ax+b)ⁿ = 1/a [(ax+b)ⁿ⁺¹/n+1]
3) ∫cos(ax+b) = 1/a sin(ax+b) + c
4) ∫sin(ax+b) = -1/a cos(ax+b) + c
5) ∫1/ax+b = 1/a ln |ax+b|+ c

Question 4

Use substitution to find integrals

Answer 4

• Let one of the expressions be u
• Let the other expression be du/dx
• ∫[du/dx] dx = du
• Subsequently, ∫u du
• Integrate and sub u to find final integral