Power Rule for Integration
This topic covers reversing the process of differentiation for expressions involving powers of x. 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.