An Integral is a function, F, which can be used to calculate the area bound by the graph of the derivative function, the x-axis, the vertical lines x=a and x=b. It is commonly written in the following form:

Int_a->b_f(x)

where,

Int is the operation for integrate

a and b represent the vertical lines bounding the area

f(x) is the derivative function

Thus for the simple function y=3x^2, we can integrate in the following way:

Y = 3x^2

F = int(3x^2)

F = (3x^3)/3

F = x^3

If the x values were 1 and 2, then the area under the curve would be:

F(2)-F(1) = 8-1 units.

However, finding the derivatives is usually never as easy as the above example. Instead there are many different methods to find the integral of complex functions. For example, one can integrate by parts to find the integral of a product of functions. Consider the following function:

f(x) = x*e^2x

We can split the function into two parts:

u = x

dv = e^2x

Integrating by parts has the following formula:

int_u*dv=u*v-int_v*du

Thus we can calculate:

du = 1

v = (1/2)*e^2x

Therefore, the integral is:

Int_x*e^2x = x*(1/2)*e^2x – int_(1/2)*e^2x*(1)

Int_x*e^2x = x*(1/2)*e^2x – int_(1/4)*e^2x + c

From this example it can be seen that finding the integral is not always straightforward. Thus, understanding the complexities and the multitude of different rules to integrate a function is crucial for the study of calculus.