Integration: Fundamentals and Applications

Integration is the mathematical process of finding the antiderivative of a function. In the JAMB syllabus, it is treated as the inverse of differentiation. Integration is essentially the process of summing up small parts to find a whole, such as the area under a curve.

1. Basic Rules of Integration (Algebraic Functions)

For any algebraic function $x^n$, the power rule for integration is defined as:

$$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$

Where $n \neq -1$ and $C$ is the constant of integration. The constant $C$ is necessary because the derivative of any constant is zero, so we must account for it when reversing the process.

• Lineary Property: $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$
• Scalar Multiplication: $\int kf(x) dx = k \int f(x) dx$

2. Integration of Trigonometric Functions

Candidates are expected to know the standard integrals for simple trigonometric functions. These are derived directly from the reverse of differentiation:

• $\int \sin x dx = -\cos x + C$
• $\int \cos x dx = \sin x + C$
• $\int \sec^2 x dx = \tan x + C$
• For functions with a coefficient, like $\sin(ax)$, the result is: $\int \sin(ax) dx = -\frac{1}{a}\cos(ax) + C$

3. Definite Integrals

A definite integral has upper and lower limits ($a$ and $b$). It represents a specific numerical value rather than a general function. The Fundamental Theorem of Calculus states:

$$\int_{a}^{b} f(x) dx = [F(x)]_{a}^{b} = F(b) - F(a)$$

4. Area Under the Curve

One of the primary applications of integration in JAMB is calculating the area bounded by a curve $y = f(x)$, the x-axis, and the vertical lines $x = a$ and $x = b$.

Formula: Area = $\int_{a}^{b} y dx$

If the area is below the x-axis, the integral will yield a negative value; however, area is always expressed as a positive magnitude.