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

Example 1 Find $\displaystyle \int{2e^x}dx$. $\displaystyle \int{2e^x}dx = 2e^x +c$ Example 2 Find $\displaystyle \int{e^{2x+1}}dx$. $\displaystyle \int{e^{2x+1}}dx = \dfrac{1}{2}e^{2x+1} + c$ Example 3 Find $\displaystyle \int{\dfrac{1}{e^x}}dx$. \( \begin{align} \displaystyle\int{\dfrac{1}{e^x}}dx &= \int{e^{-x}}dx \\&= \dfrac{1}{-1}e^{-x} +c \\&= -\dfrac{1}{e^x} +c\end{align} \) Example 4 Find $\displaystyle \int{\sqrt{e^x}}dx$. \( \begin{align} \displaystyle\int{\sqrt{e^x}}dx &= \int{(e^x)^{\frac{1}{2}}}dx \\&= \int{e^{\frac{x}{2}}}dx \\&= \dfrac{1}{2}e^{\frac{x}{2}} +c \\&= \dfrac{1}{2}\sqrt{e^x} […]