Question

Utilize métodos de integração para resolver a seguinte integral:

34861ad96791fb869e7931565cda8d75.docx

Answer 1

$\mathrm{u=\ln{x}\ \to\ u'=\dfrac{1}{x}\ \ \|\ \ v'=\sqrt{x}=x^{\frac{1}{2}}\ \to\ v=\dfrac{2}{3}x^{\frac{3}{2}}}}\\\\ \mathrm{\int\sqrt{x}\ln{x}\ dx=\dfrac{2}{3}x^{\frac{3}{2}}\ln{x}-\int\dfrac{1}{x}\dfrac{2}{3}x^{\frac{3}{2}}\ dx=}\\\\ \mathrm{=\dfrac{2}{3}x^{\frac{3}{2}}\ln{x}-\dfrac{2}{3}\int x^{\frac{1}{2}}\ dx=\dfrac{2}{3}x^{\frac{3}{2}}\ln{x}-\dfrac{2}{3}\dfrac{2}{3}x^{\frac{3}{2}}=}\\\\\ \mathrm{=\dfrac{2}{3}x^{\frac{3}{2}}\bigg(\dfrac{3\ln{x}-2}{3}\bigg)=\boxed{\mathrm{\dfrac{2x\sqrt{x}(3\ln{x}-2)}{9}}}}$

$\mathrm{\int_1^4\sqrt{x}\ln{x}\ dx=\bigg(\mathrm{\dfrac{2x\sqrt{x}(3\ln{x}-2)}{9}}\bigg)\bigg|_1^4=}\\\\ \mathrm{\dfrac{2.4\sqrt{4}(3\ln{4}-2)}{9}-\dfrac{2.1\sqrt{1}(3\ln{1}-2)}{9}=}\\\\ \mathrm{=\dfrac{16(6\ln{2}-2)}{9}-\dfrac{2(0-2)}{9}=\dfrac{32(3\ln{2}-1)+4}{9}=}\\\\ \mathrm{=\dfrac{4}{9}\bigg(8(3\ln{2}-1)+1\bigg)=\boxed{\mathrm{\dfrac{4}{9}\bigg(24\ln{2}-7\bigg)}}}}$