Integration SS3 Further Mathematics Lesson Note

SPECIFIC OBJECTIVES: At the end of the lesson, students should be able to; 

1. Understand integration as the reverse process of differentiation; 
2. Integrate algebraic polynomials including 1 𝑥 and logarithmic function; 
3. Apply integration to kinematics problems including velocity time graphs; 
4. Use the definite integral to calculate the area under a curve 

INSTRUCTIONAL RESOURCES: Charts showing integration as the reverse process of differentiation etc. 

LESSON PRESENTATION: The teacher presents the lesson step by step by first asking the students questions based on previous lessons, for example, what is differentiation, rules of differentiation, Etc. 

STEP 1 

MODE: Entire Class 

TEACHER’S ACTIVITIES 

Definition: 

The process of reversing differentiation is called INTEGRATION or integration is the opposite of differentiation INDEFINITE INTEGRATION Integration is represented by elongated s i.e. ∫ In general ∫ 𝑥 𝑛 𝑑𝑥 = 𝑥 𝑛+1 𝑛+1 + 𝐾 I.e. increase the power by one and divide by new power. Where K is a constant of the integration 

Example: Evaluate ∫ 5𝑥 4 𝑑𝑥 

Solution 

∫ 5𝑥 4 𝑑𝑥 = 

5𝑥 4+1 4 + 1 + 𝐾 = 

5𝑥 5 5 + 𝐾 = 𝑥 5 + k

I.e. increase the power by one and divide by new power. Where K is a constant of the integration 

Example: 

Evaluate ∫ 5𝑥 4 𝑑𝑥 Solution ∫ 5𝑥 4 𝑑𝑥 = 5𝑥 4+1 4 + 1 + 𝐾 = 5𝑥 5 5 + 𝐾 = 𝑥 5 + 𝐾 

STUDENTS ACTIVITIES 

1. Evaluate these indefinite integrals 

• ∫ 2𝑥 4 𝑑𝑥 
• ∫(𝑥 3 + 4𝑥) 𝑑𝑥 
• ∫(3𝑥 2 − 2𝑥 − 11) 𝑑𝑥 
•  ∫(𝑡 3 − 2𝑡 2 )𝑑𝑡 
• ∫ ( 1 2 𝑥 2 + 1 3 𝑥) 𝑑x
• ∫(0.1𝑥 4 − 0.2𝑥 3) 𝑑𝑥 

2. Simplify the following indefinite integrals 

• ∫ 1 𝑥 2 𝑑𝑥 
• ∫ 1 𝑥 𝑑𝑥
• ∫ ( 𝑥 3+𝑥 2−2𝑥+1 𝑥 3 ) 𝑑𝑥 
• ∫ ( 𝑥 2+3𝑥+2 𝑥+1 ) 𝑑𝑥
• 𝑖𝑓 𝑑𝑦 𝑑𝑥 = 𝑥 3 − 2𝑥 2 + 3, find y 

3. Evaluate these integrals 

• ∫ 𝑥(3𝑥 + 2) 𝑑𝑥
• ∫ 𝑥(𝑥 3 + 4𝑥) 𝑑𝑥
• ∫ 𝑡 2 (4𝑡 − 6) 𝑑𝑡
• ∫(𝑥 2 − 4) 2 𝑑𝑥 
• ∫ 𝑥 2 (𝑥 2 − 4) 2 𝑑𝑥
• ∫ 2𝑝(3 − 𝑝 2 ) 2𝑑𝑝 

STEP II 

Exploration; facts about the lesson objectives using the resources around 

MODE: Entire Class 

TEACHER’S ACTIVITIES There are some basic integrals need to know; 

• ∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐾 
• ∫ sin𝑥 𝑑𝑥 = − cos 𝑥 + 𝐾 
• ∫ 𝑠𝑒𝑐2𝑥 𝑑𝑥 = ∫ 1 𝑐𝑜𝑠2𝑥 𝑑𝑥 = tan 𝑥 + k
• ∫ 1 𝑥 𝑑𝑥 = ln 𝑥 + 𝐾 v. ∫ 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝐾 

STUDENTS ACTIVITIES 

Evaluate these indefinite integrals 

• ∫(sin 𝜃 − 2 cos 𝜃) 𝑑𝜃
• ∫( 3 𝑥 + 2𝑒 𝑥 + 3 5 sin 𝑥) 𝑑𝑥
•  ∫ ( cos 𝜃 3 − sin 𝜃 3 ) 𝑑𝜃 

STEP III 

Discussion of change of variable 

MODE: Entire Class 

TEACHER’S ACTIVITIES 

INDEFINITE INTEGRALS II (CHANGE OF VARIABLE) In some problems, the integral do not fit in directly, in such cases, a substitution involving a change of the variable will help 

Example: Solve ∫ cos 2𝜃 𝑑𝜃 

Solution 

Let 𝑢 = 2𝜃 𝑑𝑢 𝑑𝜃 = 2 𝑑𝜃 = 𝑑𝑢 2 Then, substituting for 2𝜃 𝑎𝑛𝑑 𝑑𝜃 ∫ cos 2𝜃 𝑑𝜃 = ∫ cos 𝑢 𝑑𝑢 2 = 1 2 ∫ cos 𝑢 𝑑𝑢 = 1 2 sin 𝑢 + 𝐾 = 1 2 sin 2𝜃 + K

STUDENTS ACTIVITIES 

Evaluate these indefinite 

• ∫ cos 4𝑥 𝑑𝑥
• ∫ 8 cos 4𝑥 𝑑𝑥
• ∫ sin4𝑥 𝑑𝑥
• ∫ 16 sin 2𝑥 𝑑𝑥
• ∫ 8 cos 1 6 𝑥 𝑑𝑥
• ∫( 2 3 cos 4𝑥 − 3 sin 1 4 𝑥) 𝑑𝑥(𝑣) ∫ 3 (3𝑥 − 1) 1 2𝑑𝑥(𝑣𝑖) ∫ 2 (2𝑥 − 3) 2 3𝑑𝑥
• ∫ 𝑒 5𝑥 𝑑𝑥 
•  ∫(2 sin(−3𝜃) + 4 cos(− 1 3 𝜃))𝑑𝜃(𝑖𝑥) ∫ 2𝑒 −𝑥 𝑑𝑥(𝑥) ∫ 1 3 𝑒 1 3 𝑥𝑑𝑥 

STEP IV 

Discussion of the numerator is a derivative of the denominator 

MODE: Entire Class 

TEACHER’S ACTIVITIES 

If the numerator is a derivative of the denominator That is,∫ 𝑔 1𝑥 𝑔(𝑥) 𝑑𝑥 = ln 𝑔(𝑥) + 𝐾 Where 𝑔 1𝑥 is the derivative of 𝑔(𝑥) 

Example: Evaluate ∫ 2 1+2𝑥 𝑑𝑥 

STUDENTS ACTIVITIES 

Evaluate 

• ∫ 7𝑥 3(𝑥 2−3) 𝑑𝑥 
• ∫ 𝑥 3 2𝑥 4+1 𝑑𝑥 

Evaluation: The teacher will evaluate students based on this lesson; for example, the integrals of ∫𝐾𝑥 𝑛 𝑑𝑥 are given as? Etc. 

ASSIGNMENT 

Evaluate 

• ∫(2𝑐𝑜𝑠 1 3 𝜃 − 4 sin(−𝑥)) 𝑑𝜃
• ∫( 1 5 cos (− 1 10 𝜃) + sin 3𝜃) 𝑑𝜃 

REFERENCES 

• Further mathematics for SS3 by P.N Lassa and S.A Ilori 
• Hidden Facts in Further Mathematics page 332 
• Excellence in Mathematics page 156
• Further Mathematics Project 3