Standard +0.8 This is a two-part Further Maths question requiring (a) integration by substitution with exponentials, then (b) solving a first-order linear ODE using integrating factor method where the integrating factor itself requires the result from part (a). The connection between parts and the need to recognize that the integrating factor leads back to the preliminary integral elevates this above a routine FP2 question, though the individual techniques are standard.
6. (a) Using the substitution \(t = x ^ { 2 }\), or otherwise, find
$$\int x ^ { 3 } \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x$$
(b) Find the general solution of the differential equation
$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = x \mathrm { e } ^ { - x ^ { 2 } } , \quad x > 0$$
6. (a) Using the substitution $t = x ^ { 2 }$, or otherwise, find
$$\int x ^ { 3 } \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x$$
(b) Find the general solution of the differential equation
$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = x \mathrm { e } ^ { - x ^ { 2 } } , \quad x > 0$$
\hfill \mbox{\textit{Edexcel FP2 2003 Q6 [10]}}