Preview
IB DP Mathematics: Applications and Interpretation | HL | Calculus — Topic 5.11
📋 Learning Intentions
- Find indefinite and definite integrals of trigonometric functions \(\sin(ax+b)\), \(\cos(ax+b)\), and \(\frac{1}{\cos^2(ax+b)}\)
- Find indefinite and definite integrals of exponential functions of the form \(e^{ax+b}\) and \(\frac{1}{ax+b}\)
- Apply integration by substitution to reduce composite integrals to standard forms
- Calculate definite integrals using substitution, correctly transforming limits of integration
- Interpret the result of a definite integral in context — area under a curve, accumulated change, or total quantity
- Verify an integration result by differentiating the answer and confirming it matches the original integrand
Real-World Introduction
Electrical engineers model alternating current (AC) as a sinusoidal function of time. To find the total charge delivered over a time interval, they must integrate \(\sin\) or \(\cos\) functions — sometimes with linear arguments like \(\sin(100\pi t)\). Without the ability to integrate these forms, power calculations in every household circuit would be impossible.
Exponential functions model radioactive decay, population growth, and cooling. The total amount of a substance that decays over a period is found by integrating \(e^{-kt}\). Logarithmic integrals arise whenever a rate is inversely proportional to a quantity — for example, the time taken to drain a tank through a small hole. These are not exotic scenarios: they are the standard tools of applied science and engineering.
🌍 Why This Matters In signal processing, the energy carried by a sound wave over one cycle is computed as \(\int_0^{T} A^2\sin^2(\omega t)\,dt\). Evaluating this integral requires both trigonometric integration and substitution — the two core skills of this topic.
Core Concepts & Definitions
Every integral rule is the reverse of a differentiation rule. Before tackling composite functions, it is essential to know the standard results for \(\sin x\), \(\cos x\), \(e^x\), and \(\frac{1}{x}\) — these appear directly in the IB formula booklet and form the foundation for all further work.
📖 Key Definition — Standard Integral Forms (IB Formula Booklet)
- \(\displaystyle\int \sin x \, dx = -\cos x + C\)
- \(\displaystyle\int \cos x \, dx = \sin x + C\)
- \(\displaystyle\int \frac{1}{\cos^2 x} \, dx = \int \sec^2 x \, dx = \tan x + C\)
- \(\displaystyle\int e^x \, dx = e^x + C\)
- \(\displaystyle\int \frac{1}{x} \, dx = \ln|x| + C \quad (x \neq 0)\)
These results are given in the IB Mathematics AI HL