Calculation of primitives and applications
Derivatives of composite functions
Recognising a composite form
Some functions are calculated using an inner function u. If u is differentiable on I, the following forms must be identified:
| Form of f | A primitive F | Condition |
|---|---|---|
| f = u' * u^n | F = u^(n+1)/(n+1) | n an integer, n ≥ 1 |
| f = u'/u | F = ln(u) | u(x) > 0 |
| f = u' * e^u | F = e^u | none |
| f = u'/sqrt(u) | F = 2*sqrt(u) | u(x) > 0 |
| f = u' * cos(u) | F = sin(u) | none |
| f = u' * sin(u) | F = -cos(u) | none |
Example 1
f(x) = 2x * (x²+3)⁴. Let u(x) = x² + 3, so u'(x) = 2x. We recognise the form u' * u^n where n = 4. F(x) = u⁵/5 = (x² + 3)⁵ / 5.
Example 2
f(x) = 2x/(x²+1). Let u(x) = x²+1 > 0, so u'(x) = 2x. This is the form u'/u. F(x) = ln(x² + 1).
Method
- Identify u(x) and calculate u'(x).
- Check that the factor in front corresponds to u'(x), up to a multiplicative constant.
- Adjust using linearity if a coefficient is missing: for f(x) = x*(x²+1)⁴, we write x = (1/2)*2x, so F(x) = (1/2) * (x²+1)⁵/5.
Common pitfall
The most common pitfall: forgetting to check that u’ is actually present in front of u^n or u/... Many pupils apply the formula without checking this coefficient, which results in an incorrect antiderivative. Always differentiate the result you have found to check it.