What values of can I use the power rule for in P1 integration?

The power rule applies for all rational values of (integers, fractions, negative numbers) except . Integration of is covered in P3 and will not appear in P1 integration questions.

Integration (Edexcel IAL Pure Mathematics 1)

Edexcel International A-Level Mathematics· P1 Sections 5.1-5.2· 15 min read

1. Indefinite Integration as Reverse Differentiation★★☆☆☆⏱ 4 min

✓ Calculator OK

Integration is the inverse operation of differentiation. If you differentiate a function $f (x)$ to get its derivative $f^{'} (x)$ , integrating $f^{'} (x)$ will return you to the original function plus an arbitrary constant, known as the constant of integration.

📘 Definition

Indefinite Integral

$\int f (x) d x$

The antiderivative of the integrand function $f (x)$ , written as $\int f^{'} (x) d x = f (x) + C$ , where $C$ is the constant of integration.

📐 Worked Example

Given that $\frac{d}{d x} (5 x^{3} - 2 x) = 15 x^{2} - 2$ , write the indefinite integral of $15 x^{2} - 2$ with respect to $x$ .

1
Recognize that integration reverses differentiation, so we start with the original function that was differentiated.
2
Add the arbitrary constant $C$ to account for any constant term that would have been eliminated during differentiation.
3
$\int (15x^2 - 2) dx = 5x^3 - 2x + C$

Exam tip:

Always add the constant $C$ immediately after writing the result of an indefinite integral — this is a mandatory mark for every P1 integration question.

2. Power Rule for Integration of $x^n$★★★☆☆⏱ 5 min

✓ Calculator OK

The power rule for integration is derived directly from the power rule for differentiation. For any rational number $n$ where $n \neq = - 1$ , the rule is as follows:

\int ax^n dx = a \cdot \frac{x^{n+1}}{n+1} + C

You can integrate sums and differences of terms by integrating each term individually, just as you do when differentiating. Always simplify terms into the form $a x^{n}$ before integrating to avoid errors.

📐 Worked Example

Find $\int (8 x^{3} + 6 x^{1/2} - 4 x^{- 3}) d x$ .

1
Integrate each term separately using the power rule:
2
$\int 8x^3 dx = 8 \cdot \frac{x^4}{4} = 2x^4$
3
$\int 6x^{1/2} dx = 6 \cdot \frac{x^{3/2}}{3/2} = 6 \cdot \frac{2}{3}x^{3/2} = 4x^{3/2}$
4
$\int -4x^{-3} dx = -4 \cdot \frac{x^{-2}}{-2} = 2x^{-2}$
5
Combine all integrated terms and add the constant $C$ for the final result:
6
$= 2x^4 + 4x^{3/2} + 2x^{-2} + C$

3. Calculating the Constant of Integration★★★☆☆⏱ 4 min

✓ Calculator OK

The constant $C$ is only arbitrary if no additional information about the function is provided. If you are given a point $(x, y)$ that lies on the curve of the original function, you can substitute these values into the integrated function to solve for the exact value of $C$ .

📐 Worked Example

Given that $\frac{d y}{d x} = 6 x - 4$ , and the curve $y = f (x)$ passes through the point $(1, 5)$ , find the full equation of $y = f (x)$ .

1
First integrate the derivative to get the general form of $y$ :
2
$y = \int (6x -4) dx = 3x^2 -4x + C$
3
Substitute the given values $x = 1$ , $y = 5$ into the equation to solve for $C$ :
4
$5 = 3(1)^2 -4(1) + C$
5
$5 = 3 - 4 + C$
6
$5 = -1 + C \implies C = 6$
7
Write the final equation with the calculated value of $C$ :
8
$y = 3x^2 -4x +6$

Exam tip:

Verify your answer by differentiating your final function to confirm you get the original derivative given in the question — this takes seconds and catches most careless errors.

4. Common Pitfalls

Wrong move:

Forgetting to add the constant $C$ when writing indefinite integrals.

Why:

Constant terms disappear during differentiation, so you must account for all possible original constant terms. This is one of the most common mark-loss errors in P1 integration questions.

Correct move:

Write $+ C$ immediately after finishing integration of all terms, before moving to any other part of the question.

Wrong move:

Adding 1 to the power but forgetting to divide by the new power $n + 1$ .

Why:

Confusion between the integration power rule and differentiation power rule (where you multiply by the original power).

Correct move:

Memorize the integration power rule mnemonic: 'raise power by 1, divide by the new power, add $C$ '.

Wrong move:

Trying to integrate $x^{- 1} = 1/ x$ using the power rule in P1 questions.

Why:

The power rule only applies for $n \neq = - 1$ , and integration of $1/ x$ is not part of the P1 specification.

Correct move:

If you encounter $1/ x$ in a P1 integration question, you have made an error simplifying terms earlier; go back and check your algebraic manipulation.

Wrong move:

Integrating products or quotients of terms as separate terms without expanding first, e.g. integrating $\int x (x + 3) d x$ as $\frac{x ^{2}}{2} (\frac{x ^{2}}{2} + 3 x) + C$ .

Why:

The power rule only applies to individual terms in the form $a x^{n}$ , not products or quotients of terms.

Correct move:

Always expand brackets and simplify all terms into separate $a x^{n}$ terms before integrating.

Wrong move:

Making arithmetic errors with negative powers, e.g. integrating $x^{- 4}$ to $\frac{x ^{- 3}}{3}$ instead of $\frac{x ^{- 3}}{- 3}$ .

Why:

Mental calculation errors when working with negative exponents.

Correct move:

Write every step of the power rule calculation explicitly when working with negative or fractional powers, do not simplify in your head.

5. Quick Reference Cheatsheet

Rule	Formula	Exam Notes
Reverse differentiation	$\int f^{'} (x) d x = f (x) + C$	Always include $+ C$ for all indefinite integrals
Power rule	$\int a x^{n} d x = a \frac{x ^{n + 1}}{n + 1} + C$	Valid for all rational $n \neq = - 1$
Sum/difference rule	$\int [f (x) \pm g (x)] d x = \int f (x) d x \pm \int g (x) d x$	Integrate each term separately
Find constant $C$	Substitute $(x, y)$ point into integrated function	Use when given a point on the original curve

6. Frequently Asked

Do I need to learn definite integration for P1 exams?

No, definite integration, area under curves, and the trapezium rule are all part of the P2 specification, not P1. You only need to study indefinite integration for P1 assessments.

What values of $n$ can I use the power rule for in P1 integration?

The power rule applies for all rational values of $n$ (integers, fractions, negative numbers) except $n = - 1$ . Integration of $x^{- 1} = 1/ x$ is covered in P3 and will not appear in P1 integration questions.

Going deeper

official_documentEdexcel IAL Maths 2018 SpecificationRefer to P1 Section 5 for full integration requirements

What's Next

Now that you have mastered P1 indefinite integration, you are prepared for more advanced integration topics in later Edexcel IAL units. In P2, you will learn definite integration, which enables you to calculate the area under a curve between two bounds, as well as the trapezium rule for approximating integral values. In P3, you will extend your skills to integrate trigonometric functions, exponential functions, and the reciprocal function $1/ x$ . For P1 exams, focus on practicing integration of integer, fractional, and negative powers of $x$ , and solving for the constant $C$ using given points, as these make up 100% of P1 integration questions. Always show full working to gain all method marks, even if you use a calculator to verify your final answer.