# Definite Integral of Exponential Functions

\large \begin{align} \displaystyle \int_{n}^{m}{e^{ax+b}}dx &= \dfrac{1}{a}\big[e^{ax+b}\big]_{n}^{m} \\ &= \dfrac{1}{a}\big[e^{am+b}-e^{an+b}\big] \\ \end{align}

### Example 1

Find $\displaystyle \int_{2}^{4}{e^{2x-4}}dx$, leaving the answer in exact form.

\begin{align} \displaystyle \int_{2}^{4}{e^{2x-4}}dx &= \dfrac{1}{2}\big[e^{2x-4}\big]_{2}^{4} \\ &= \dfrac{1}{2}\big[e^{2 \times 4-4}-e^{2 \times 2-4}\big] \\ &= \dfrac{e^4-e^{0}}{2} \\ &= \dfrac{e^4-1}{2} \end{align}

### Example 2

Find $\displaystyle \int_{0}^{1}{(e^x+1)^2}dx$.

\begin{align} \displaystyle \int_{0}^{1}{(e^{2x} + 2e^x + 1)}dx &= \big[\dfrac{1}{2}e^{2x} + 2e^x + x\big]_{0}^{1} \\ &= \big[\dfrac{1}{2}e^{2 \times 1} + 2e^1 + 1\big]-\big[\dfrac{1}{2}e^{2 \times 0} + 2e^0 + 0\big] \\ &= \big[\dfrac{1}{2}e^2 + 2e + 1\big]-\big[\dfrac{1}{2} \times 1 + 2 \times 1\big] \\ &= \dfrac{1}{2}e^2 + 2e + 1-\dfrac{1}{2}-2 \\ &= \dfrac{1}{2}e^2 + 2e-\dfrac{3}{2} \end{align}

### Example 3

Find $\displaystyle \int_{0}^{4}{\dfrac{1}{e^x}}dx$.

\begin{align} \displaystyle \int_{0}^{4}{\dfrac{1}{e^x}}dx &= \int_{0}^{4}{e^{-x}}dx \\ &= -\big[e^{-x}\big]_{0}^{4} \\ &= -\big[e^{-4}-e^{-0}\big] \\ &= -\dfrac{1}{e^4}+1 \end{align}

### Example 4

Find $\displaystyle \int_{1}^{4}{e^{1-x}}dx$.

\begin{align} \displaystyle \int_{1}^{4}{e^{1-x}}dx &= -\big[e^{1-x}\big]_{1}^{4} \\ &= -\big[e^{1-4}-e^{1-1}\big] \\ &= -\big[e^{-3}-e^{0}\big] \\ &= -\big[\dfrac{1}{e^3}-1\big] \\ &= – \dfrac{1}{e^3}+1 \end{align}