DIFFERENTIATION & INTEGRATION Chapter 4 DIFFERENTIATION & INTEGRATION

OUTLINE OF CHAPTER 4: DIFFERENTIATION & INTEGRATION Integration by Substitution Integration by Parts Integration by Tabular Method Integration by Partial Fraction Differentiation Derivative of power function (power rule) Derivative of a constant times a function Derivative of sum and difference rules Product rule Quotient rule Derivative of trigonometric functions Derivative of exponential and logarithmic functions Chain rule Implicit Differentiation

4.2.1 DERIVATIVE of power rule Example 4.1: Exercise 4.1:

4.2.2 DERIVATIVE of a constant times a function Example 4.2: Exercise 4.2:

4.2.3 DERIVATIVE of sum & difference rules Example 4.3: Exercise 4.3:

Exercise 1: power rule

Exercise 1: answer

4.2.4 the product rules Example 4.4: Exercise 4.4:

Exercise 2: product rule

Exercise 2: answer

4.2.5 the quotient rules Example 4.5: Exercise 4.5:

Exercise 3: quotient rule

Exercise 3: answer

Exercise 4:

4.2.6 DERIVATIVE of trigonometric functions

Example 4.6: Exercise 4.6:

Exercise 5: trigonometric functions

4.2.7 DERIVATIVE of logarithmic functions

Example 4.7: Exercise 4.7:

Properties of ln:

4.2.8 the chain rules Example 4.8: Exercise 4.8:

Example: chain rule

Exercise 6:

Exercise 6: answer

Conclusion (differentiation) Power Rule Product Rule Quotient Rule Chain Rule

4.2.9 implicit differentiation Implicit differentiation is the process of taking the derivative when y is defined implicitly or in others y is a function of x. STEP 1: Differentiate both side with respect to x. STEP 2: Collect dy/dx terms on the left hand side of the equation STEP 3: Solve for dy/dx

Example 4.9: Exercise 4.9:

Exercise 7:

Exercise 8

4.3 integration Integration is the inverse process of differentiation process. Derivative formula Equivalent integral formula

Exercise 9:

4.3.1 indefinite integral The constant factor k can be taken out from an integral, The integral of a sum or difference equals the sum or difference of the integral, that is

Example 4.12: Exercise 4.11:

Exercise 10:

4.3.2 definite integral If f(x) is a real-valued continuous function on closed interval [a,b] and F(x) is an indefinite integral of f(x) on [a,b], then

Basic Properties of Definite Integrals:

Example 4.13: Exercise 4.12:

Exercise 11:

4.4 technique of integration Integration by Substitution Integration by Parts Integration by Tabular Method Integration by Partial Fraction

4.4.1 integration by substitution Step 1: Choose appropriate u, Step 2: Compute Step 3: Substitute and in the integral Step 4: Evaluate the integral in term of u Step 5: Replace , so that the final answer will be in term of x

Example:

Example 4.14: Exercise 4.13:

Example:

Exercise 12:

4.4.2 integration by parts Involve products of algebraic and transcendental functions. For example: The formula: A priority order to choose u:

4.4.2 integration by parts Step 1: Choose the appropriate u and dv. (Note: the expression dv must contain dx) Step 2: Differentiate u to obtain du and integrate dv to obtain v. (Note: Do not include the constant C when integrating dv since we are still in the process of integrating) Step 3: Substitute u, du, v and dv into formula and complete the integration. (Note: Remember to include the constant C in the final answer)

Example 4.15: Exercise 4.13:

Exercise 13:

4.4.3 integration by tabular methods The formula: Note: Can be used to evaluate complex integrations especially repeated integrations (when u=xn). Step 1: u can be differentiated repeatedly with respect to x until becoming zero. Step 2: v’ can be integrated repeatedly with respect to x.

Example:

Exercise 14:

4.4.4 integration by partial fraction Consider a function, where P and Q are polynomials and Q(x)≠ 0 Case I (improper fraction): If the deg P(x) ≥ deg Q(x), long division is applied to obtain remainder Example:

Exercise 15:

Case II (proper fraction): If the deg P(x) < deg Q(x), factorize the denominator (Q(x)) into its prime factors. i) Linear factor ii) Linear factor

Case II (proper fraction): If the deg R(x) < deg T(x), factorize the denominator (T(x)) into its prime factors. iii) Quadratic factor iv) Quadratic factor

Example 4.16: Exercise 4.15:

Example:

Example:

Exercise 16:

Exercise: