Worked example 1
Integrate a composite power
Find the integral of 6(2x + 1)³ with respect to x.
- Increase the power to 4.
- Divide the coefficient 6 by 2 × 4.
- Add the constant of integration.
Answer: (2x + 1)⁴ + C
Higher Mathematics
Higher Maths Integration of shifted powers
Practise integration of shifted powers for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Integrate (x + q)ⁿ and (px + q)ⁿ with the correct inner-factor adjustment.
Before you start
- Integrate powers.
- Recognise a linear expression inside a power.
- Check an integral by differentiating.
Explanation
A power of px + q can be integrated without expansion by reversing the chain rule. Increase the outer power and divide by both the new power and p.
This method applies for n not equal to −1.
Method and rules
- ∫(px + q)n dx = + C
- The rule requires n ≠ −1.
- Differentiate the answer to check the factor p cancels.
Worked examples
Worked example 2
Method check
Integrate 3(3x − 2)⁴
- Divide 3 by 3 × 5.
So: (3x − 2)^ + C
Watch out
- Dividing by n + 1 but not by the inner coefficient p.
- Using the rule when n = −1
- Forgetting + C for an indefinite integral.
Exam reminder
For integration of shifted powers, show the defining equation or formula before simplifying. Check that every final value satisfies the original restrictions, interval or context.
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