Question

determine a integral de f(x) = 5x/(x² + 3) no intervalo de [-1 , 1]

Answer 1

Resolução da questão, vejamos:

Resolver a integral definida:

$\mathsf{\displaystyle\int_{-1}^{1}~\dfrac{5x}{x^{2}+3}}~\mathsf{dx}}$

Suponhamos que:

u = x² + 3 = > du = 2x dx;

$\mathsf{xdx=\dfrac{du}{2}}}$

Vejamos:

$\mathsf{\displaystyle\int~\dfrac{5x~dx}{x^{2}+3}}~=\mathsf{5\displaystyle\int~\dfrac{xdx}{x^{2}+3}}\\\\\\\ \mathsf{5\displaystyle\int~\dfrac{du}{2}~\cdot~\dfrac{1}{u}}\\\\\\\ \mathsf{\dfrac{5}{2}\displaystyle\int~\dfrac{du}{u}}\\\\\\\ \mathsf{\dfrac{5}{2}~ln(u)}}$

Substituindo u por x² + 3, teremos:

$\mathsf{\dfrac{5}{2}~ln(u)}}=\mathsf{\dfrac{5}{2}~ln(x^{2}+3)}}$

Definindo os limites de integração, temos:

$\mathsf{\displaystyle\int_{-1}^{1}~\dfrac{5x}{x^{2}+3}}~\mathsf{dx}}=\mathsf{\dfrac{5}{2}~ln(x^{2}+3)}\bigg|_{-1}^{1}}\\\\\\\ \mathsf{\dfrac{5}{2}~ln(1^{2}+3)-\dfrac{5}{2}~ln((-1)^{2}+3)}}\\\\\\\ \mathsf{\dfrac{5}{2}~ln(4)-\dfrac{5}{2}~ln(4)}}=\Large\boxed{\boxed{\mathbf{0.}}}}}}}}}$

Assim sendo, podemos afirmar que:

$\mathsf{\displaystyle\int_{-1}^{1}~\dfrac{5x}{x^{2}+3}}~\mathsf{dx}}=0.$

Espero que te ajude (^.^)