Worked example 1
Integrate a polynomial
Find ∫(6x² − 4x + 3) dx.
- Increase each power by 1.
- Divide by the new power.
- Integrate the constant as 3x.
- Add C.
Answer: 2x³ − 2x² + 3x + C
Higher Mathematics
Higher Maths Indefinite integration
Practise indefinite integration for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Reverse the power rule for algebraic expressions and include the constant of integration.
Before you start
- Differentiate powers confidently.
- Understand that integration reverses differentiation.
- Remember that a family of functions needs + C.
Explanation
Indefinite integration reverses the power rule. Increase the power by 1, divide by the new power and add C.
The exception n = −1 is usually handled separately, so avoid using the power rule there.
Method and rules
- ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C, n ≠ −1
- ∫axⁿ dx = axⁿ⁺¹/(n + 1) + C
- ∫k dx = kx + C
Worked examples
Worked example 2
Check by differentiating
Check ∫4x³ dx = x⁴ + C
- Differentiate x⁴ + C
- d/dx(x⁴) = 4x³ and d/dx(C) = 0
So: The derivative returns 4x³.
Worked example 3
Integrate a negative power
Find ∫6x⁻² dx.
- Increase −2 to −1.
- Divide by −1.
Answer: -6x⁻¹ + C
Watch out
- Differentiating instead of integrating.
- Forgetting + C.
- Dividing by the old power instead of the new power.
Exam reminder
For indefinite integrals, + C is part of the answer. You only find a numerical C if a condition is supplied.
Related Higher Maths topics
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