# Integration of Exponential Functions

Home > Integration The base formula of integrating exponential function is obtained from deriving $e^x$. \begin{align} \displaystyle \dfrac{d}{dx}e^x &= e^x \\ e^x &= \int{e^x}dx \\ \therefore \int{e^x}dx &= e^x +c \\ \end{align} This base formula is extended to the following general formula.  \begin{align} \displaystyle \dfrac{d}{dx}e^{ax+b} &= e^{ax+b} \times \dfrac{d}{dx}(ax+b) \\ &= e^{ax+b} [...]

# Basic Integration Rules

Antiderivatives In many cases in calculus, it is known that the rate of change of one variable with respect to another, but we do not have a formula which relates the variables. In other words, it is known that $\dfrac{dy}{dx}$, but we need to know $y$ in terms of $x$. The process of finding $y$ [...]

# Calculation of Areas under Curves

Consider the function $f(x)=x^2+2$. We wish to estimate the green area enclosed by $y=f(x)$, the $x$-axis, and the vertical lines $x=1$ and $x=4$. Suppose we divide the $x$-interval into three strips of width 1 unit. Upper Rectangles The diagram below shows upper rectangles, which are rectangles with top edges at the maximum value of the [...]