Choosing an Integration Method
An integration method is chosen from the visible structure, not from the length of the expression. Start with the least disruptive step that makes the integrand simpler.
Scan before you calculate
Visible cue
A familiar function with a linear inside
Method
Direct reverse differentiation
Keep the exponential unchanged and divide by the derivative 3 of its exponent.
- Simplify algebra and use a direct reverse derivative or a required trigonometric identity first.
- Check for a numerator proportional to a denominator derivative, or for the exact form .
- If the question gives a substitution, use it and rewrite the whole integral. Choosing a general substitution is not required in this syllabus.
- Decompose a suitable rational function into partial fractions. Use by parts for a product only when differentiating one factor makes the next integral easier.
Common mistake
Multiplication does not automatically mean by parts. For example, a supplied substitution may turn a product into one power in the new variable.
Allow one method to reveal another
A question can require more than one method. After each step, inspect the new integral again instead of continuing the first method by habit.
Worked example
A short match hidden by the numerator
The denominator derivative is . The numerator is half of it, so partial fractions and by parts would both add unnecessary work.
Answer
Key idea
The best method is the one whose first valid step creates a standard integral with less work.
Verify the structure and the final answer
- Before integrating, name the feature that justifies the method.
- After integrating, differentiate the answer and simplify until it matches the original integrand.
- For a definite integral, also check the sign and likely size from the graph or from the integrand on the interval.
Your turnIndependent method check [6 marks]
State the first useful method for each integral.
Show worked answer
- Inverse tangent, with .
- Derivative over function.
- Integration by parts, with .