To integrate a power of x, add one to the index and divide by the new index, then add a constant of integration: ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + c, for any n except −1. Roots and fractions must be rewritten as powers of x first, and products expanded, because there is no simple reverse product rule.
The integration power rule
- ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + c, for n ≠ −1
- ∫ axⁿ dx = axⁿ⁺¹/(n + 1) + c
- Integrate a sum term by term
- ∫ k dx = kx + c for a constant k
Rewrite before integrating
- √x = x^(1/2) and 1/√x = x^(−1/2)
- 1/xᵏ = x^(−k)
How do you integrate a power of x?
Integration reverses differentiation. For a power of x, add one to the index and then divide by the new index. Constant multipliers stay in front, and a sum is integrated term by term. For example, 12x³ becomes 12x⁴/4 = 3x⁴.
An indefinite integral always ends with + c. Leaving it out costs a mark.
Negative and fractional indices
The rule works for any rational index except −1, but each term must first be written as a power of x: √x = x^(1/2), 1/x² = x^(−2). Take care adding one to the index: −3 + 1 = −2, and ½ + 1 = 3/2. Dividing by a fraction such as 3/2 is the same as multiplying by 2/3.
Rewrite products and quotients first
There is no simple reverse product or quotient rule at this stage, so expand any product and split any fraction into separate powers of x before integrating. You cannot integrate a fraction by integrating its top and bottom separately.
Why + c, and why n = −1 is excluded
Differentiating any constant gives 0, so infinitely many functions share the same derivative, differing only by a constant. That whole family is written with + c.
The rule fails for n = −1, because adding one would give an index of 0 and a division by zero. The integral of x⁻¹ = 1/x is ln|x| + c, which you meet in Pure 3.
Integrating with letters as well as numbers
Exam questions sometimes leave constants as letters, such as ∫ (ax² + bx) dx. Integrate exactly as before, treating a and b as numbers: the result is (a/3)x³ + (b/2)x² + c. Comparing this with a given answer, term by term, lets you find the constants.
Checking an integral by differentiating
Because integration reverses differentiation, you can always check your answer: differentiate it and you should get back the original expression. If you do not, an index or coefficient has gone wrong, usually in the dividing step. In “show that” questions, set out the rewriting clearly and reach the printed result exactly.
Worked examples
Example 1. Find ∫ (12x³ − 2x) dx.
- 12x³: the index goes from 3 to 4, then divide by 4, giving 3x⁴.
- −2x: the index goes from 1 to 2, then divide by 2, giving −x².
- Add the constant of integration.
Answer: 3x⁴ − x² + c.
Marks: M1 for raising an index and dividing, A1 for 3x⁴, A1 for −x² + c.
Example 2. Find ∫ (√x + 1/x²) dx.
- Rewrite: ∫ (x^(1/2) + x^(−2)) dx.
- x^(1/2) integrates to x^(3/2)/(3/2) = ⅔x^(3/2).
- x^(−2) integrates to x^(−1)/(−1) = −x^(−1).
Answer: ⅔x^(3/2) − 1/x + c.
Marks: M1 for rewriting, A1 for each term.
Example 3. Find ∫ x(2x − 3) dx.
- Expand: ∫ (2x² − 3x) dx.
- Integrate term by term.
Answer: ⅔x³ − (3/2)x² + c.
Marks: M1 for expanding, A1 for the answer.
Example 4. Given that ∫ ax² dx = 3x³ + c, find a.
- ∫ ax² dx = (a/3)x³ + c.
- Compare with 3x³: a/3 = 3.
Answer: a = 9.
Marks: M1 for (a/3)x³, A1 for a = 9.
Example 5. Find ∫ (x + 2)(x − 2) dx and check your answer.
- Expand: x² − 4.
- Integrate: x³/3 − 4x + c.
- Check: differentiating x³/3 − 4x gives x² − 4.
Answer: x³/3 − 4x + c.
Marks: M1 for expanding, A1 for the integral.
Common mistakes
- Forgetting + c. Every indefinite integral needs a constant of integration.
- Multiplying by the new index instead of dividing. x³ integrates to x⁴/4, not 4x⁴.
- Errors with negative indices. x^(−2) integrates to x^(−1)/(−1); the new index is −1, not −3.
- Integrating a fraction top and bottom. Split it into separate powers of x first.
- Using the rule for 1/x. The power rule does not work for n = −1.
How integrating powers of x is examined
Integration begins in Pure 1 (WMA11) of Edexcel International A-Level, and in the first year of Edexcel A-Level (9MA0). Questions on this skill ask you to:
- write an expression as a sum of powers of x, then integrate it
- integrate terms with negative and fractional indices
- find unknown constants by comparing an integral with a given result
- show that an integral equals a printed answer
Practise with the Pure 1 integration exam questions, which include full mark schemes, and the Pure 1 integration worksheet. The A-Level integration revision engine gives unlimited further practice.
Next: use + c to find a specific curve in indefinite integrals. Integration reverses basic differentiation.
Frequently asked questions
What is the integral of xⁿ?
xⁿ⁺¹/(n + 1) + c, for any n except −1.
Why do you add + c when integrating?
Because the derivative of any constant is zero, so the original function could have had any constant term.
How do you integrate √x?
Write it as x^(1/2). Adding one to the index gives x^(3/2), and dividing by 3/2 gives ⅔x^(3/2) + c.