Integral Calculus
Integration as an Inverse Process of Differentiation
In differential calculus, we are given a function and we have to find its derivative. In integral calculus, we are given the derivative of a function and we are asked to find the original function, also known as its anti-derivative or primitive.
For example, if we know that the derivative of x² is 2x, then an anti-derivative of 2x is x².
The process of finding anti-derivatives is called integration.
∫ f(x) dx = F(x) + CSince the derivative of any constant is zero, if F(x) is an anti-derivative of f(x), then F(x) + C is also an anti-derivative, where C is any arbitrary constant. This is called the constant of integration.
Visualizing Integration as Area
One of the most powerful applications of integration is calculating the area under a curve. We can approximate this area by dividing it into many thin rectangles and summing their areas. As the number of rectangles approaches infinity, this approximation becomes exact.
Basic Integration Rules
| Rule Name | Formula |
|---|---|
| Power Rule for Integrals | ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, n ≠ -1 |
| Integral of 1/x | ∫ (1/x) dx = ln|x| + C |
| Integral of eˣ | ∫ eˣ dx = eˣ + C |