Integration

Summary

Integration is the process of finding the original function from its derivative, known as the anti-derivative or integral. It involves adding an arbitrary constant 'C' to account for the constant lost during differentiation.

• Integration of Exponential Functions — The reverse process of differentiation for exponential functions. Example: $\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C$
• Integration of 1/(ax+b) — The integral of a function of the form $\frac{1}{ax+b}$. Example: $\int \frac{1}{ax+b} \, dx = \frac{1}{a} \ln|ax+b| + C$
• Integration of sin(ax + b), cos(ax + b), and sec²(ax + b) — Involves reversing the differentiation of these trigonometric functions. Example: $\int \sin(ax+b) \, dx = -\frac{1}{a} \cos(ax+b) + C$
• The Trapezium Rule — A method to approximate the area under a curve by dividing it into trapezoids. Example: $\text{Area} = \frac{h}{2}(y_0 + 2y_1 + 2y_2 + ... + 2y_{n-1} + y_n)$
• Integration by Substitution — Simplifies integration by changing variables to match standard forms. Example: $\int \sqrt{x} \, dx$ using substitution $u = \sqrt{x}$
• Integration by Parts — Used when integration by substitution is not feasible, based on the product rule of differentiation. Example: $\int u \, dv = uv - \int v \, du$
• Partial Fractions in Integration — Decomposes rational functions into simpler fractions for easier integration. Example: $\int \frac{1}{x^2 - 1} \, dx$ using partial fractions.

Exam Tips

Key Definitions to Remember

• Integration is the reverse process of differentiation.
• An indefinite integral includes an arbitrary constant 'C'.

Common Confusions

• Forgetting to add the constant 'C' in indefinite integrals.
• Mixing up integration by parts and substitution methods.

Typical Exam Questions

• What is the integral of $e^{2x}$? $\frac{1}{2} e^{2x} + C$
• How do you integrate $\sin(3x)$? $-\frac{1}{3} \cos(3x) + C$
• Use the trapezium rule to estimate $\int_0^2 x^2 \, dx$. Approximate using $\frac{h}{2}(y_0 + 2y_1 + y_2)$

What Examiners Usually Test

• Ability to apply integration techniques to various functions.
• Understanding of when to use substitution or parts.
• Application of the trapezium rule for definite integrals.