AS Level
Mathematics 9709 P1

9. Integration

Written by: Tharun Athreya
Formatted by: Tharun Athreya

If \( \frac{dy}{dx} = x^n \), then
\( y = \frac{1}{n+1} x^{n+1} + c \)
(where \( c \) is an arbitrary constant and \( n \neq -1 \)).
‘Increase the power \( n \) by 1 to obtain the new power, then divide by the new power. Remember to add a constant \( c \) at the end.’
The special symbol \( \int \) is used to denote integration.
Integral operator is \( dx \).
How do I find \( c \)?
- STEP 1: Rewrite the function into a more easily integrable form
  - Each term needs to be a power of \( x \) (or a constant).
- STEP 2: Integrate each term and remember “+c”
  - Increase power by 1 and divide by new power.
- STEP 3: Substitute the coordinates of a given point to form an equation in \( c \).
  - Solve the equation to find \( c \).

The general rule: If \( n \neq -1 \) and \( a \neq 0 \), then
\(\int (ax + b)^n \,dx = \frac{1}{a(n+1)} (ax + b)^{n+1} + c \).

You can use this for complicated expressions.
General rule: If \( \frac{d}{dx} [F(x)] = f(x) \), then
\(\int f(x) \,dx = F(x) + c\).

General Rule: \(\int_{a}^{b} f(x) \,dx = [F(x)]_{a}^{b} = F(b) – F(a)\),
where:
- \( a \) – lower limit
- \( b \) – upper limit
Note: The ‘c’s cancel out to make the process simple.
How do I find a definite integral?
- STEP 1: If not given a name, call the integral. (It could save time)
- STEP 2: If necessary, rewrite the integral into a more easily integrable form.
- STEP 3: Integrate without applying the limits.
- STEP 4: Substitute the limits into the function and calculate the answer.
Note: You can also use a calculator to check the answer. (Advanced scientific calculators can compute definite integrals)

🔥 Exam Tip

Look out for questions that ask you to find an indefinite integral in one part (so “+c” needed), then in a later part use the same integral as a definite integral (where “+c” is not needed).

Area under a curve is the area bounded by:
- The x-axis
- The graph of y = f(x)
- The vertical line x = a
- The vertical line x = b
How do I find the area under a curve?
- General rule: If y = f(x) is a function with y ≥ 0, then the area, A, bounded by the curve y = f(x), the x-axis, and the lines x = a and x = b is given by the formula:
  A = ∫[a,b] y dx
  - If the area lies underneath the x-axis, the value of the integral will be negative. Since an area cannot be negative, take the magnitude only.
- Area enclosed by a curve and the y-axis: If x = f(y) is a function with x ≥ 0, then the area, A, bounded by the curve x = f(y), the y-axis, and the lines y = a and y = b is given by:
  A = ∫[a,b] x dy (when x ≥ 0).
- Area bounded by a curve and a line or by two curves:
  A = ∫[a,b] f(x) dx – ∫[a,b] g(x) dx
  (or)
  A = ∫[a,b] [f(x) – g(x)] dx