This equation shows how to find the integral of a polynomial term. It is often used in conjunction with the Addition Rule for Integration to take the integral of a polynomial.
| \( x \) | This is a symbol for any generic variable. It can hold any value, whether that be an integer or a real number, or a complex number, or a matrix etc. |
| \( k \) | This symbol represents any given integer, \( k \in \htmlClass{sdt-0000000122}{\mathbb{Z}}\). |
| \( \int \) | This is the symbol for an integral, sometimes referred to as an antiderivative. Graphically, it can be understood as the area between a curve and the axis the integral is taken with respect to. |
| \( C \) | This symbol represents the constant of integration. It must be added to the result of all definite integrals to encompass all possible solutions that satisfy the integral. |
| \( \:d \) | This is the symbol for a differential. It represents an infinitesimally small (infinitely close to zero) change in whatever variable it is with respect to. |
| \( a \) | This is a symbol for any generic constant. It can hold any numerical value |
Consider the power rule for differentiation:
\[\frac{\htmlClass{sdt-0000000102}{\:d}}{\htmlClass{sdt-0000000102}{\:d} \htmlClass{sdt-0000000003}{x}}(\htmlClass{sdt-0000000121}{a} \htmlClass{sdt-0000000003}{x}^{\htmlClass{sdt-0000000015}{k}}) = \htmlClass{sdt-0000000015}{k} \htmlClass{sdt-0000000121}{a} \htmlClass{sdt-0000000003}{x}^{\htmlClass{sdt-0000000015}{k} - 1}\]
Also consider the definition of an indefinite integral:
\[\htmlClass{sdt-0000000060}{\int} \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}) \htmlClass{sdt-0000000102}{\:d} \htmlClass{sdt-0000000003}{x} = (\htmlClass{sdt-0000000101}{\lim}_{\htmlClass{sdt-0000000104}{n} \to \htmlClass{sdt-0000000108}{\infty}}\htmlClass{sdt-0000000080}{\sum}_{\htmlClass{sdt-0000000018}{i} = 1}^{\htmlClass{sdt-0000000104}{n}} \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}_{\htmlClass{sdt-0000000018}{i}}) \htmlClass{sdt-0000000105}{\Delta} \htmlClass{sdt-0000000003}{x}_{\htmlClass{sdt-0000000018}{i}}) + \htmlClass{sdt-0000000070}{C}\]
More coming soon...
Consider \(\htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}) = 4 \htmlClass{sdt-0000000003}{x}\), what is \(\htmlClass{sdt-0000000060}{\int} \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}) \htmlClass{sdt-0000000102}{\:d} \htmlClass{sdt-0000000003}{x}\)?
\(\htmlClass{sdt-0000000121}{a} = 4\)
\(\htmlClass{sdt-0000000015}{k} = 1\)
We can now plug these values in to arrive at our:
\[ \htmlClass{sdt-0000000060}{\int} \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}) \htmlClass{sdt-0000000102}{\:d} \htmlClass{sdt-0000000003}{x} = \frac{4}{1 + 1}\htmlClass{sdt-0000000003}{x}^{1 + 1} = 2\htmlClass{sdt-0000000003}{x}^2 \]