Question

usando o método de integração por partes,e lembrando que (lnx)=1/x, determine
∫(lnx).2x dx ∫u.dv=u.v-∫v.du

a) x².lnx-x²/2+c
b)1/x.2+c
c) x².lnx+c
d)x².lnx+x²/2+c

Answer 1

Olá

$\displaystyle \mathsf{\int \ell n(x)\cdot 2xdx}$

Tira a constante de 'dentro' da integral

$\displaystyle \mathsf{2\int \ell n(x)\cdot xdx}$

Integração por partes

$\displaystyle \boxed{\mathsf{\int udv~=~u\cdot v~-~\int vdu}}$

Escolhe o 'u' e o 'dv' de acordo com a regra do LIATE

$\displaystyle \mathsf{u=\ell nx\qquad\qquad\qquad\qquad dv=xdx}\\\\\mathsf{du= \frac{1}{x}dx \qquad\qquad\qquad\qquad v~=~\int dv= \frac{x^2}{2} }$

Substituindo na fórmula

$\displaystyle \mathsf{\int \ell n(x)2xdx=I}\\\\\\\\\mathsf{I=2\left( \frac{x^2}{2}\cdot \ell n(x)~-~\int \frac{x^2}{2}\cdot \frac{1}{x}dx \right) }$

Simplificando

$\displaystyle \mathsf{I=2\left( \frac{x^2}{2}\cdot \ell n(x)~-~\int \frac{x^{\diagup\!\!\!\!2}}{2}\cdot \frac{1}{\diagup\!\!\!\!x} dx\right) }\\\\\\\\\mathsf{I=2\left( \frac{x^2}{2}\cdot \ell n(x)~-~\int \frac{x}{2}dx \right) }\\\\\\\mathsf{I=2\left( \frac{x^2}{2}\cdot \ell n(x) ~-~ \frac{1}{2} \cdot \frac{x^2}{2}+C \right)}\\\\\\\mathsf{I=2\left( \frac{x^2}{2}\cdot \ell n(x)~- ~ \frac{x^2}{4}+C\right) }$

Fazendo a distributiva do 2

$\displaystyle \mathsf{I=2\cdot \frac{x^2}{2}\cdot \ell n(x)~-~2\cdot \frac{x^2}{4}+C }$

Simplificando

$\displaystyle \mathsf{I=\diagup\!\!\!\!2\cdot \frac{x^2}{\diagup\!\!\!\!2}\cdot \ell n(x)~-~\diagup\!\!\!\!2\cdot \frac{x^2}{\diagup\!\!\!\!\!4} +C}\\\\\\\boxed{\mathsf{I=x^2\cdot \ell n(x)~- ~ \frac{x^2}{2}+C }}~\qquad\qquad\qquad \Longrightarrow\qquad\text{Letra A)}$