### Product Rule & Integration by Parts

Fluency exercise

Can you find the missing positive numbers $a$ to $d$?
1. $\displaystyle{\int_1^a \dfrac{1}{3x} \,dx = \ln 2}$
2. $\displaystyle{\int_0^3 ax e^{2x} \,dx = be^6 + 2}$
3. $\displaystyle{\int_{1}^{3} \dfrac{cx}{\sqrt{25-x^2}} \,dx = 4\sqrt{6} - 4c}$
4. $\displaystyle{\int_{\frac{\pi}{2}}^{\frac{b}{a}\pi} x^2\sin cx \,dx = \dfrac{\pi d - \pi^2 + c}{a}}$