Which expression is equal to \( \int s^2 e^{7x} dx? \) A - HSC - SSCE Mathematics Extension 2 - Question 2 - 2021 - Paper 1

Integration by parts formula is given by ( \int u dv = uv - \int v du ). In this case, we can choose ( u = s^2 ) and ( dv = e^{7x} dx ). This leads to ( du = 2s ds ) and ( v = \frac{1}{7} e^{7x} ).

Applying the integration by parts:

[ \int s^2 e^{7x} dx = \frac{1}{7} s^2 e^{7x} - \frac{2s}{7} \int s e^{7x} ds ]

This form can aid in determining which expression corresponds.

After evaluating the options, it is clear that the correct choice is:

[ A: \frac{1}{7} e^{7x} - \frac{5}{7} \int s^4 e^{7x} dx ]

This aligns with the integration performed.