Integration – SS3 Further Mathematics Lesson Note (Second Term)

Integration is the reverse process of differentiation. It is written with the elongated S sign (∫). To integrate xⁿ, increase the power by one, divide by the new power and add K, the constant of integration. This lesson also covers the basic trigonometric and exponential integrals, change of variable, and integrals whose numerator is the derivative of the denominator.

Behavioural objectives

By 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 the 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, and similar materials.

Lesson presentation

The teacher presents the lesson step by step. Begin by asking the students questions based on previous lessons, for example: What is differentiation? What are the rules of differentiation?

Step 1: What is indefinite integration?

Mode: Entire class

Teacher’s activities

The process of reversing differentiation is called integration. Integration is the opposite of differentiation. It is represented by an elongated S, that is, ∫.

In general, ∫xⁿ dx = xⁿ⁺¹/(n+1) + K, where n ≠ -1. That is, increase the power by one and divide by the new power. K is the constant of integration.

Example: Evaluate ∫5x⁴ dx.

Solution

∫5x⁴ dx = 5x⁴⁺¹/(4+1) + K = 5x⁵/5 + K = x⁵ + K

Students’ activities

1. Evaluate these indefinite integrals:
   • ∫2x⁴ dx
   • ∫(x³ + 4x) dx
   • ∫(3x² – 2x – 11) dx
   • ∫(t³ – 2t²) dt
   • ∫(½x² + ⅓x) dx
   • ∫(0.1x⁴ – 0.2x³) dx
2. Simplify the following indefinite integrals:
   • ∫(1/x²) dx
   • ∫(1/x) dx
   • ∫((x³ + x² – 2x + 1)/x³) dx
   • ∫((x² + 3x + 2)/(x + 1)) dx
   • If dy/dx = x³ – 2x² + 3, find y.
3. Evaluate these integrals:
   • ∫x(3x + 2) dx
   • ∫x(x³ + 4x) dx
   • ∫t²(4t – 6) dt
   • ∫(x² – 4)² dx
   • ∫x²(x² – 4)² dx
   • ∫2p(3 – p²)² dp

Step 2: What are the basic trigonometric and exponential integrals?

Mode: Entire class. Students use the facts and resources around them to study the lesson objectives.

Teacher’s activities

There are some basic integrals students need to know:

• ∫cos x dx = sin x + K
• ∫sin x dx = -cos x + K
• ∫sec²x dx = ∫(1/cos²x) dx = tan x + K
• ∫(1/x) dx = ln x + K
• ∫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 3: How does change of variable work?

Mode: Entire class

Teacher’s activities

In some problems the integral does not fit a basic form directly. In such cases, a substitution that changes the variable will help.

Example: Solve ∫cos 2θ dθ.

Solution

Let u = 2θ. Then du/dθ = 2, so dθ = du/2.

Substituting for 2θ and dθ:

∫cos 2θ dθ = ∫cos u (du/2) = ½∫cos u du = ½ sin u + K = ½ sin 2θ + K

Students’ activities

Evaluate these indefinite integrals:

1. ∫cos 4x dx
2. ∫8 cos 4x dx
3. ∫sin 4x dx
4. ∫16 sin 2x dx
5. ∫8 cos(x/6) dx
6. ∫((2/3) cos 4x – 3 sin(x/4)) dx
7. ∫3(3x – 1)^1/2 dx
8. ∫2(2x – 3)^2/3 dx
9. ∫e^5x dx
10. ∫(2 sin(-3θ) + 4 cos(-θ/3)) dθ
11. ∫2e^-x dx
12. ∫(1/3) e^x/3 dx

Step 4: What if the numerator is the derivative of the denominator?

Mode: Entire class

Teacher’s activities

If the numerator is the derivative of the denominator, that is, ∫(g′(x)/g(x)) dx = ln g(x) + K, where g′(x) is the derivative of g(x).

Example: Evaluate ∫(2/(1 + 2x)) dx.

Students’ activities

Evaluate:

• ∫ 7𝑥 3(𝑥 2−3) 𝑑𝑥
• ∫(x³/(2x⁴ + 1)) dx

Evaluation

The teacher evaluates the students on the lesson. Sample questions:

1. How is the integral ∫Kxⁿ dx given?
2. What is the constant K called in an indefinite integral?
3. State the integrals of cos x, sin x and e^x.
4. Evaluate ∫cos 4x dx using a change of variable.

Assignment

Evaluate:

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