#1**+2 **

Using the substitution \(u = x + 3\), \(du = dx\), we have the change of integration boundary \(\begin{cases}x = 1 \implies u= 4\\x = 2 \implies u = 5\end{cases}\). Then we have \(\displaystyle\int_1^2 \dfrac{(x + 3)^2}3\,dx =\dfrac13\int_4^5u^2\,du\). You can use the power rule to integrate it. Note that \(\displaystyle\int_a^b f(x)\,dx = F(b) - F(a)\) where F is an antiderivative of f.

MaxWong Apr 19, 2022