Power Rule for Integration
Integrating powers of x reverses the process of differentiation; the key skill for ESAT is algebraically simplifying complex expressions, like expanded brackets or fractions, into a sum of simple xn terms before applying the standard integration rule.
Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The fundamental rule is to integrate `xn` by adding one to the power and dividing by the new power.
- For indefinite integrals (those without limits), you must always add a '+ c', the constant of integration.
- Many ESAT questions require you to first manipulate the expression. This often involves expanding brackets or splitting a fraction into separate terms.
- You can integrate expressions with multiple terms by integrating each term individually and then summing the results.
- The rule does not apply for n = -1 (i.e., integrating 1/x), and this case will not be tested.
- No advanced methods like integration by parts or substitution are required for the ESAT.
Formulae
∫ kxn dx = (k × x^(n+1)) / (n+1) + c To find the indefinite integral of a single term where x is raised to a rational power 'n', as long as n is not equal to -1.
Definitions
- Indefinite Integral
- The general anti-derivative of a function, which represents a family of functions. It is always written with a '+ c' (constant of integration).
- Integrand
- The expression or function that is to be integrated.
- Constant of Integration
- An arbitrary constant term 'c' added to an indefinite integral. It is necessary because the derivative of any constant is zero.
Worked example
Find the indefinite integral of (2x - 3)2 / x^(1/2).
- 1
Step 1:
Simplify the integrand before integrating.
First, expand the squared bracket in the numerator:
(2x - 3)2 = 4x2 - 12x + 9 - 2
Step 2:
Rewrite the expression as a single fraction:
(4x2 - 12x + 9) / x^(1/2).
- 3
Step 3:
Split the fraction into three separate terms by dividing each term in the numerator by the denominator, using the index law x^a / x^b = x^(a-b).
- 4
Step 4:
The terms become:
4x^(2 - 1/2) - 12x^(1 - 1/2) + 9x^(-1/2) which simplifies to 4x^(3/2) - 12x^(1/2) + 9x^(-1/2).
- 5
Step 5:
Now, integrate each term using the rule 'add one to the power, divide by the new power'.
- 6
Step 6:
For 4x^(3/2), the new power is 5/2.
The integral is 4 × x^(5/2) / (5/2) = 4 × (2/5) × x^(5/2) = (8/5)x^(5/2).
- 7
Step 7:
For -12x^(1/2), the new power is 3/2.
The integral is -12 × x^(3/2) / (3/2) = -12 × (2/3) × x^(3/2) = -8x^(3/2).
- 8
Step 8:
For 9x^(-1/2), the new power is 1/2.
The integral is 9 × x^(1/2) / (1/2) = 9 × 2 × x^(1/2) = 18x^(1/2).
- 9
Step 9:
Combine the integrated terms and add the constant of integration 'c'.
Answer: (8/5)x^(5/2) - 8x^(3/2) + 18x^(1/2) + c
Common mistakes
- ×Arithmetic errors when dealing with fractional or negative powers. For example, getting `-1/2 + 1` wrong, or mixing up division by a fraction (e.g. `n / (3/2)` should be `n × (2/3)`).
- ×Making a mistake during the initial algebraic simplification, such as incorrectly expanding brackets or misapplying index laws when dividing terms.
- ×Forgetting to include the constant of integration, `+ c`, in the final answer for an indefinite integral.
No-calculator tips
- ✓To divide by a fraction, multiply by its reciprocal. For example, when dividing by a new power of 5/2, it is much easier to think of it as multiplying by 2/5.
- ✓Before you begin, rewrite all terms in the standard form `kxn`. Convert all roots (`√(x)`, `cuberoot(x)`) and reciprocals (`1/x2`) into index notation (`x^(1/2)`, `x^(1/3)`, `x^(-2)`).
- ✓When simplifying fractions with multiple terms in the numerator, deal with each term one at a time to avoid confusion. Mentally (or on scrap paper) handle `term1/denominator`, then `term2/denominator`, etc.