Standard +0.8 This is a Further Maths question combining integration by parts with improper integrals, requiring careful handling of limits at infinity. While the integration by parts itself is standard, students must correctly set up the limit, apply the given result, and justify convergence—more demanding than typical A-level integration but routine for Further Maths FP2.
5 In this question you must show detailed reasoning.
Evaluate \(\int _ { 0 } ^ { \infty } 2 x \mathrm { e } ^ { - x } \mathrm {~d} x\). [0pt]
[You may use the result \(\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x } = 0\).]
5 In this question you must show detailed reasoning.
Evaluate $\int _ { 0 } ^ { \infty } 2 x \mathrm { e } ^ { - x } \mathrm {~d} x$.\\[0pt]
[You may use the result $\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x } = 0$.]
\hfill \mbox{\textit{OCR Further Pure Core 2 2017 Q5 [4]}}