Solutions - 0010-04

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

\begin{matrix} u = x & v^{'} = e^{4 x} \\ u^{'} = 1 & v = \frac{1}{4} e^{4 x} \end{matrix} \begin{matrix} ⟶ \int u \cdot v^{'} = \int x e^{4 x} original \\ ⟶ \int u^{'} \cdot v = \int \frac{1}{4} e^{4 x} final \end{matrix}

(2) Apply IBP formula $u v - \int u^{'} \cdot v$ and compute integral:

\begin{matrix} u v - \int u^{'} \cdot v & ≫ ≫ {\frac{x}{4} e^{4 x} |}_{0}^{3} - \int_{0}^{3} \frac{1}{4} e^{4 x} d x \\ ≫ ≫ {\frac{x}{4} e^{4 x} - \frac{1}{16} e^{4 x} |}_{0}^{3} \\ ≫ ≫ {(\frac{x}{4} - \frac{1}{16}) e^{4 x} |}_{0}^{3} \\ ≫ ≫ e^{12} (\frac{3}{4} - \frac{1}{16}) - 1 (- \frac{1}{16}) \\ ≫ ≫ \frac{11}{16} e^{12} + \frac{1}{16} \end{matrix}

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