Question 5 - A-Level Maths

Exam Board	AQA
Module	FP3 (Further Pure Mathematics 3)
Year	2012
Session	January
Topic	Integration with Partial Fractions

Explain why $\int _ { \frac { 1 } { 2 } } ^ { \infty } \frac { x ( 1 - 2 x ) } { x ^ { 2 } + 3 \mathrm { e } ^ { 4 x } } \mathrm {~d} x$ is an improper integral.
(1 mark)
By using the substitution $u = x ^ { 2 } \mathrm { e } ^ { - 4 x } + 3$, find $$\int \frac { x ( 1 - 2 x ) } { x ^ { 2 } + 3 \mathrm { e } ^ { 4 x } } \mathrm {~d} x$$
Hence evaluate $\int _ { \frac { 1 } { 2 } } ^ { \infty } \frac { x ( 1 - 2 x ) } { x ^ { 2 } + 3 \mathrm { e } ^ { 4 x } } \mathrm {~d} x$, showing the limiting process used.