Integration:

Integration is the reverse of differentiation—it's like finding your way back home after taking a journey. When you integrate a function, you're finding the original function that, when differentiated, gives you what you started with. Think of it like calculating the total distance a moving vehicle has traveled if you know its speed at each moment. A trader in Lagos market who knows exactly how much profit he makes each day can use integration to find his total profit over a whole month.

The main integration rules you need are: the power rule (add one to the power, divide by the new power), integration of constants, and the sum rule (integrate each term separately). Always remember to add the constant of integration (+ C) at the end—this constant represents all possible original functions.

When you integrate sine and cosine functions, you're finding the area under their curves. The integral of sin(x) is simply -cos(x) + C, while the integral of cos(x) is sin(x) + C. Notice how sine and cosine swap roles during integration, with a negative sign appearing for sine. This relationship comes from their connection through differentiation—they're inverse operations of each other.

Think of it like calculating the total distance a pendulum travels: integrating the sine function that describes its motion gives you meaningful information about its behaviour. These formulas appear frequently in physics problems involving waves and oscillations that you'll encounter in JAMB questions.

When solving these problems, always remember to add your constant of integration, C, because many exam questions specifically test whether you include it.

When you integrate a function, you're essentially finding the total area trapped between a curve and the x-axis. Think of it like calculating how much land you own if your property's boundary follows a curved line. To find this area, you use the definite integral, which involves finding the antiderivative of your function and then substituting your upper and lower limits.

The formula is straightforward: the area equals F(b) minus F(a), where F is your antiderivative and a and b are your boundary points. For example, if a trader's profit follows the curve y = 2x from x = 0 to x = 5, integrating this gives you the total profit earned across that period. Most JAMB questions focus on simple polynomial functions, so mastering basic integration rules is essential.