Solutions - 0010-03

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

$$\begin{array}{cc}u=\ln x& v^{\prime }=x^3\\ u^{\prime }={\displaystyle \frac{1}{x}}& v={\displaystyle \frac{x^4}{4}}\end{array}\begin{array}{c}⟶\int u\cdot v^{\prime }=\int x^3\ln x\text{original}\\ ⟶\int u^{\prime }\cdot v=\int {\displaystyle \frac{x^3}{4}}\text{final}\end{array}$$

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

$$\begin{array}{cc}uv-\int u^{\prime }\cdot v& \gg \gg \frac{x^4}{4}\ln x-\int \frac{x^3}{4}dx\\ & \\ & \gg \gg \frac{x^4}{4}\ln x-\frac{x^4}{16}+C\end{array}$$