Standard +0.8 This is an improper integral requiring integration by parts and careful handling of limits at infinity. While the technique is standard for Further Maths students and the given limit simplifies the work, it still requires proper understanding of improper integrals, correct application of integration by parts, and rigorous limit evaluation—making it moderately harder than average A-level questions but routine for FP2 students.
In this question you must show detailed reasoning.
Evaluate \(\int_0^{\infty} 2xe^{-x} dx\).
[You may use the result \(\lim_{x \to \infty} xe^{-x} = 0\).] [4]
e.g. M1 for (cid:114)2xe(cid:16)x (cid:114)(cid:179)2e(cid:16)xdx
Evaluation must be seen
Question 5:
5 | DR
(cid:16)2xe(cid:16)x (cid:14)(cid:179)2e(cid:16)xdx
Evaluation of their F(cid:62)x(cid:64) using both limits
(cid:62)0(cid:16)0(cid:64)(cid:16)(cid:62)0(cid:16)2(cid:64)
2 | *M1
A1
dep*M1
A1
[4] | 1.1
1.1
1.1
1.1 | Allow sign errors only
Must be seen
F[x](cid:32)(cid:16)2xe(cid:16)x (cid:16)2e(cid:16)x | e.g. M1 for (cid:114)2xe(cid:16)x (cid:114)(cid:179)2e(cid:16)xdx
Evaluation must be seen
In this question you must show detailed reasoning.
Evaluate $\int_0^{\infty} 2xe^{-x} dx$.
[You may use the result $\lim_{x \to \infty} xe^{-x} = 0$.] [4]
\hfill \mbox{\textit{OCR Further Pure Core 2 Q5 [4]}}