Question

Calcule a seguinte integral definida:

Answer 1

$I=\int_{1}^{2}{\mathrm{\ell n\,}x\,\mathrm{d}x}$


Encontrando a primitiva (resolver a integral indefinida) por partes:

$\int{\mathrm{\ell n\,}x\,\mathrm{d}x}\\ \\ \\ \begin{array}{cc} u=\mathrm{\ell n\,}x&\mathrm{d}v=\mathrm{d}x\\ \\ \mathrm{d}u=\dfrac{\mathrm{d}x}{x}&v=x \end{array}\\ \\ \\ \int{u \,\mathrm{d}v}=u\cdot v-\int{v \,\mathrm{d}u}\\ \\ \int{\mathrm{\ell n\,}x\,\mathrm{d}x}=x \cdot \mathrm{\ell n\,}x-x+C$


Aplicando o Teorema Fundamental do Cálculo

$I=\int_{1}^{2}{\mathrm{\ell n\,}x\,\mathrm{d}x}=\left.x \cdot \mathrm{\ell n\,}x-x\right]_{1}^{2}\\ \\ =\left(2\,\mathrm{\ell n\,}2-2 \right )-\left(1\,\mathrm{\ell n\,}1-1 \right )\\ \\ =2\,\mathrm{\ell n\,}2-2-0+1\\ \\ =2\,\mathrm{\ell n\,}2-1\\ \\ \\ \boxed{\int_{1}^{2}{\mathrm{\ell n\,}x\,\mathrm{d}x}=2\,\mathrm{\ell n\,}2-1}$