Integration

Summary

Integration is the process of finding the original function from its derivative, known as the anti-derivative or integral. It is the reverse of differentiation and includes an arbitrary constant 'C'.

• Integration of Exponential Functions — involves reversing the differentiation of exponential functions. Example: $\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C$
• Integration of 1/(ax+b) — involves using a modulus sign to account for intervals where the function does not exist. Example: $\int \frac{1}{ax+b} \, dx = \frac{1}{a} \ln|ax+b| + C$
• Integration of Trigonometric Functions — involves reversing the differentiation of functions like sin, cos, and sec². Example: $\int \sin(ax+b) \, dx = -\frac{1}{a} \cos(ax+b) + C$
• The Trapezium Rule — a method to estimate the value of a definite integral by dividing the area under a curve into trapezoids. Example: $\int_a^b f(x) \, dx \approx \frac{h}{2} [y_0 + 2(y_1 + y_2 + ... + y_{n-1}) + y_n]$
• Integration by Substitution — simplifies integrals by changing variables to match standard forms. Example: $\int \sqrt{x^2 + a^2} \, dx$ using substitution.
• Integration by Parts — used when integrals involve products of functions. Example: $\int u \, dv = uv - \int v \, du$
• Partial Fractions in Integration — decomposes rational functions into simpler fractions for 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.
• The integral of a function includes an arbitrary constant 'C'.

Common Confusions

• Forgetting to add the constant 'C' in indefinite integrals.
• Misapplying the modulus sign in integrals involving logarithms.

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 $\int_2^4 f(x) \, dx$. Apply the formula with given values.

What Examiners Usually Test

• Ability to perform integration by substitution and by parts.
• Understanding and application of the trapezium rule.
• Decomposition of functions into partial fractions for integration.