Applied Maths: Differentiation and Integration (Strand 9) — Higher Level Notes
Logarithms, standard integrals, integration by parts, the chain rule, differentiating vectors and the work done by a variable force.
1. Logarithms first, because everything else needs them
- ln a + ln b = ln ab
- ln a - ln b = ln (a/b)
- n ln a = ln an
- ln x = y is the same statement as x = ey
These are the manipulations that turn the answer to a differential equation into something you can read.
2. The standard integrals
∫xn dx = xn+1/(n+1) + c | ∫(1/x) dx = ln x + c
With a linear expression inside, divide by the coefficient of x:
∫ 1/(ax+b) dx = (1/a) ln(ax+b) + c
And the inverse-tangent form, which appears more often than students expect:
∫ 1/(x2 + a2) dx = (1/a) tan-1(x/a) + c
3. Definite integrals
Evaluate at the upper limit, subtract the value at the lower limit, and drop the constant. The order matters — reversing the limits changes the sign.
4. Integration by parts
∫u dv = uv - ∫v du
Choose u to be the factor that gets simpler when differentiated — usually the polynomial or the logarithm. If you choose the other way round the integral gets worse, which is itself a useful signal that you picked wrongly.
5. The chain rule
For a composite function, differentiate the outside and multiply by the derivative of the inside. Read in reverse, this is what makes substitution work in integration: spotting that the integrand contains a function and its own derivative is most of the technique.
6. Rates of change
The link between the calculus and the mechanics you already know:
v = ds/dt | a = dv/dt = v dv/ds
The second form of acceleration is the one to reach for when the question relates acceleration to displacement rather than to time. Recognising which form applies is usually the whole difficulty.
7. Differentiating vectors
Differentiate a vector by differentiating each component separately. If the position vector is given as a function of t, then the velocity is its derivative and the speed is the magnitude of that velocity.
Watch out — a component is not a speed
If the velocity works out as 81i - 108j, the speed is not 108. It is the magnitude, √(812 + 1082) = 135. Quoting one component as the speed is an easy mark to lose at the end of an otherwise correct question.
8. Work done by a variable force
When the force changes with position, the work is an integral rather than a product:
W = ∫ F ds
This is the bridge back to Strand 5: constant force gives W = Fs, which is just this integral when F comes outside.
Practise this chapter
QuizPerCard has Differentiation and Integration as concept cards, a formula reference and multiple-choice practice — every question with the full derivation, and an explanation of why each wrong option is wrong. Open the practice.