This equation shows that the integral of \(\htmlClass{sdt-0000000035}{e}^{\htmlClass{sdt-0000000003}{x}}\) is itself plus some constant (vertical translation)
| \( x \) | 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. |
| \( e \) | This symbol represents Euler's constant. It is approximately \(2.718\). |
| \( \int \) | 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. |
Consider the Derivative of \(\htmlClass{sdt-0000000035}{e}^{\htmlClass{sdt-0000000003}{x}}\):
\[(\htmlClass{sdt-0000000035}{e}^{\htmlClass{sdt-0000000003}{x}})\htmlClass{sdt-0000000025}{'} = \htmlClass{sdt-0000000035}{e}^{x}\]
We can now integrate both sides to get:
\[ \htmlClass{sdt-0000000060}{\int} (\htmlClass{sdt-0000000035}{e}^{\htmlClass{sdt-0000000003}{x}})\htmlClass{sdt-0000000025}{'} \htmlClass{sdt-0000000102}{\:d} \htmlClass{sdt-0000000003}{x} + \htmlClass{sdt-0000000070}{C} = \htmlClass{sdt-0000000060}{\int}(\htmlClass{sdt-0000000035}{e}^{\htmlClass{sdt-0000000003}{x}}) \htmlClass{sdt-0000000102}{\:d} \htmlClass{sdt-0000000003}{x} \]
Now also consider the integral of a derivative:
\[\htmlClass{sdt-0000000060}{\int} \htmlClass{sdt-0000000065}{f'(x)} \htmlClass{sdt-0000000102}{\:d} \htmlClass{sdt-0000000003}{x} = \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}) + \htmlClass{sdt-0000000070}{C}\]
From here we can simplify the left hand side of our second equation to get:
\[ \htmlClass{sdt-0000000035}{e}^{\htmlClass{sdt-0000000003}{x}} + \htmlClass{sdt-0000000070}{C} = \htmlClass{sdt-0000000060}{\int} \htmlClass{sdt-0000000035}{e}^{\htmlClass{sdt-0000000003}{x}} \htmlClass{sdt-0000000102}{\:d} \htmlClass{sdt-0000000003}{x} \]
By swapping the left hand side and the right hand side, we finally get:
\[ \htmlClass{sdt-0000000060}{\int} \htmlClass{sdt-0000000035}{e}^{\htmlClass{sdt-0000000003}{x}} \htmlClass{sdt-0000000102}{\:d} \htmlClass{sdt-0000000003}{x} = \htmlClass{sdt-0000000035}{e}^{\htmlClass{sdt-0000000003}{x}} + \htmlClass{sdt-0000000070}{C} \]
as required.
Coming soon...