Solutions - 0010-01

(1) Select $u$ and $v^{\prime }$ considering LIATE:

$$\begin{array}{cc}u=2x+9& v^{\prime }=e^x\\ u^{\prime }=2& v=e^x\end{array}\begin{array}{c}⟶\int u\cdot v^{\prime }\text{original}\\ ⟶\int u^{\prime }\cdot v\text{final}\end{array}$$

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(2) Apply IBP formula $uv-\int u^{\prime }\cdot v$:

$$\begin{array}{cccc}& uv-\int u^{\prime }\cdot v& \gg \gg (2x+9)e^x-\int 2e^{x}dx& \\ & & & \\ & & \gg \gg (2x+9)e^x-(2e^x+C)& \\ & & & \\ & & \gg \gg (2x+7)e^x+C& \text{(Note A)}\end{array}$$

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Note A: We can change notation $-C$ to $+C$ because the value of $C$ is arbitrary.