Introduction to Integral Calculus - SS3 Mathematics Lesson Note

Recall that in Differential Calculus we learnt how to find the differential coefficient of \(y\) i.e., \(\frac{dy}{dx}\) when given \(y\ \)as a function of \(x\). In Integral Calculus, we now wish to find back the function \(y\) when its differential coefficient \(\frac{dy}{dx}\) is given. The reverse process of finding \(y\) when \(\frac{dy}{dx}\) is given is known as Integration.

For instance, given that \(\frac{dy}{dx} = 2x\), we know that \(y = x^{2}\). On closer inspection, there are many possible solutions such as \(y = x^{2} + 2\), \(y = x^{2} - 6\) and so on, whose derivative can also be \(2x\). Indeed, any general solution of the form \(x^{2} + C\) (where \(C\) is a constant) will serve as a solution. We call \(x^{2} + C\) the integral of \(2x\) with respect to \(x\) written,

\[\int_{}^{}{2x}\ dx = x^{2} + C\]

The symbol at the beginning is the symbol for integration called the Integral. In general,

\[where\frac{dy}{dx} = g(x),\ then\ \int_{}^{}{g(x)}\ dx = y + C\]

Where \(C\) is a constant called the arbitrary constant of Integration. Provided with additional information, this constant can be determined more precisely. \(g(x)\) is the Integrand and the solution \(y + C\) is the Integral. \(dx\) shows the integration is with respect to \(x\).