Calculus – Integration

Integration

Lesson Objectives

Define integration as the reverse of differentiation
Compute basic indefinite and definite integrals
Apply standard rules of integration
Integrate basic trigonometric functions
Evaluate definite integrals over specified intervals
Solve problems involving area under a curve using integration

Lesson Introduction

Integration is the reverse process of differentiation and is a fundamental concept in calculus. It allows us to find areas under curves, total accumulated quantities, and solve many physical problems.

Lesson Content

Indefinite Integrals

An indefinite integral is written as $\int f(x)\,dx$ and represents a family of functions whose derivative is $f(x)$ . The result includes a constant of integration $C$ .

Example 1:
Find

\int x^2 \, dx

= \frac{x^3}{3} + C

Example 2:
Find

\int (3x^2 + 2x + 1)\, dx

= x^3 + x^2 + x + C

Basic Rules of Integration

Some standard rules:

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$ , $n \neq -1$
$\int \frac{1}{x} dx = \ln|x| + C$
$\int a\,dx = ax + C$

Example 3:
Find

\int \frac{1}{x} dx

= \ln|x| + C

Example 4:
Evaluate

\int (4x^3 - 2x + 7) dx

= x^4 - x^2 + 7x + C

Integration of Trigonometric Functions

Common integrals include:

$\int \sin x\,dx = -\cos x + C$
$\int \cos x\,dx = \sin x + C$

Example 5:
Find

\int \cos x\,dx

= \sin x + C

Example 6:
Evaluate

\int (\sin x + \cos x)\, dx

= -\cos x + \sin x + C

Definite Integrals

$\int_a^b f(x) dx = F(b) - F(a)$ where $F(x)$ is the antiderivative.

Example 7:
Evaluate

\int_0^3 x^2 dx

= \left[\frac{x^3}{3}\right]_0^3 = \frac{27}{3} - 0 = 9

Example 8:
Compute

\int_1^2 (3x + 2)\, dx

= \left[\frac{3x^2}{2} + 2x \right]_1^2 = \left(\frac{12}{2} + 4\right) - \left(\frac{3}{2} + 2\right) = 10 - \frac{7}{2} = \frac{13}{2}

Area Under a Curve

Use $\int_a^b f(x)\,dx$ for area between a curve and x-axis.

Example 9:
Find the area under

y = x^2

from

x = 1

x = 3

= \left[\frac{x^3}{3}\right]_1^3 = \frac{27 - 1}{3} = \frac{26}{3}

Example 10:
Area under

y = 2x + 1

from

x = 0

x = 4

= \int_0^4 (2x + 1)\,dx = \left[x^2 + x\right]_0^4 = (16 + 4) - 0 = 20

Exercises

[WAEC] Evaluate $\int_1^4 (2x + 3)\,dx$ [Past Question]
Find $\int (4x^3 - 6x^2 + 2x - 1)\, dx$
[NECO] Evaluate $\int \frac{1}{x} dx$ [Past Question]
Compute $\int \cos x\, dx$
Find $\int_0^{\pi/2} \sin x\, dx$
[WASSCE] What is $\int_2^5 x\, dx$ ? [Past Question]
Evaluate $\int_1^2 3x^2 dx$
Find the area under the curve $y = 3x + 1$ from $x = 0$ to $x = 2$
[WAEC] Determine $\int_1^3 (x^2 - 2x + 1)\,dx$ [Past Question]
Find $\int (5\sin x - 2\cos x)\, dx$

Conclusion/Recap

This lesson introduced integration as the reverse of differentiation. You learned to compute indefinite and definite integrals, apply integration rules, integrate trigonometric functions, and use integration to calculate areas under curves. Mastery of these techniques is essential for solving real-world and advanced calculus problems.