Question

Use o método de integração por parte para determinar a integral indefinida

∫x³㏑x dx

Answer 1

$\int_{}{} x^2 ln(x) \, dx \\\\ Use \: \int_{}{} u \, dv = uv - \int_{}{} v \, du \\\\ u \rightarrow ln(x) \\ dv \rightarrow x^2 \, dx \\ du \rightarrow u \, dx \\ v \rightarrow \int_{}{} 1 \, dv \\\\ ln(x) \times {x^3 \over 3} - \int_{}{} {x^3 \over 3} \times { 1 \over x} \, dx \\\\ {ln(x)x^3 \over 3} - {1 \over 3} \int_{}{} x^2 \, dx \\\\ {ln(x)x^3 \over 3} - {1 \over 3} \times { x^3 \over 3} \\\\ { 3ln(x) x^3 - x^3 \over 9 } \\\\ \boxed{ { x^3(3ln(x) -1) \over 9} + C{,} \: C \in \mathbb{R}}$