Integration 3

Summary

Integration involves finding the antiderivative of a function, which is the reverse process of differentiation.

• Integrating standard functions — involves finding the integral of basic functions like polynomials, exponentials, and trigonometric functions.
  Example: The integral of $e^{ax}$ is $\frac{1}{a}e^{ax} + C$.
• Trigonometric identities — used to simplify trigonometric functions before integrating. Example: $\sin^2(x)$ can be rewritten using the identity $\sin^2(x) = \frac{1 - \cos(2x)}{2}$.
• Reverse chain rule — a method for integrating composite functions by reversing the chain rule of differentiation. Example: Integrate $f'(g(x))g'(x)$ to get $f(g(x)) + C$.

Exam Tips

Key Definitions to Remember

• Integration is the reverse process of differentiation.
• The integral of $e^{ax}$ is $\frac{1}{a}e^{ax} + C$.

Common Confusions

• Forgetting to add the constant of integration $C$.
• Mixing up the rules for differentiation and integration.

Typical Exam Questions

• What is the integral of $\sin(x)$? Answer: $-\cos(x) + C$
• How do you integrate $\cos^2(x)$? Answer: Use the identity $\cos^2(x) = \frac{1 + \cos(2x)}{2}$ and integrate.
• What is the integral of $f'(g(x))g'(x)$? Answer: $f(g(x)) + C$

What Examiners Usually Test

• Ability to integrate standard functions and apply trigonometric identities.
• Use of the reverse chain rule in integration problems.