Chapter 13 - Section 13.4 - The Definite Integral: Algebraic Viewpoint and the Fundamental Theorem of Calculus - Exercises - Page 998: 30

$\dfrac{3}{2} \ln (3)$

Given: $I=\int_{0}^{2} \dfrac{3x}{x^2+2}\ dx$ Let us consider that $u=x^2+2 \implies dx=\dfrac{du}{2x}$ Now, we have $I= \int_{0}^{2} \dfrac{3x}{u} (\dfrac{du}{2x})$ or, $=\dfrac{3}{2} \int_{0}^{2} \dfrac{1}{u} \ du$ or, $=\dfrac{3}{2} [\ln (u)]_0^2 +C$ or, $=\dfrac{3}{2} [\ln (x^2+2)]_0^2 +C$ or, $=\dfrac{3}{2} [\ln (2^2+2)]-\dfrac{3}{2} [\ln (0^2+2)]$ or, $=\dfrac{3}{2} \ln (3)$