Challenging +1.2 This improper integral requires recognizing the substitution u = √x (or direct integration using power rules), evaluating limits at infinity, and handling the improper nature carefully. While it involves multiple steps and requires proper treatment of the infinite limit, the algebraic manipulation is straightforward once the substitution is made, and the convergence is clear. It's moderately harder than average due to the improper integral aspect and need for detailed reasoning, but doesn't require particularly deep insight or novel techniques.
6 In this question you must show detailed reasoning.
Determine the exact value of \(\int _ { 9 } ^ { \infty } \frac { 18 } { x ^ { 2 } \sqrt { x } } \mathrm {~d} x\).
Correct use of 9 as a lower limit and any letter (except x) for the upper
limit (so must be considering a finite upper limit) in their integrated
expression (indicated by their power increased by 1). Need not see
mention of limiting process for this mark.
32
−
Taking limit as 𝑘 → ∞ for their expression of the form a x (so
1
=0oe is B0). Implied by, for example,
3
2 − 32
lk i m→ − k − ... = 0 − ... but not, for example, for
3
2 − 3
− k 2 −...=0−... without clear use of limiting process.
3
cao from correct integrated expression and finite upper limit (so
dependent on both previous M marks but not the B mark). Accept
equivalent exact forms e.g. 12.
27
Question 6:
6 | DR
18 2 − 3
dx=18− x 2(+c)
x2 x 3
2 − 3 2 − 3
=18lim− k 2 −− 9 2
k→ 3 3
32
−
k → 0 as k →
4
=
9 | M1*
M1
B1dep*
A1
[4] | 1.1
1.1
2.1
2.2a | 32
−
For obtaining a x where a0.
Correct use of 9 as a lower limit and any letter (except x) for the upper
limit (so must be considering a finite upper limit) in their integrated
expression (indicated by their power increased by 1). Need not see
mention of limiting process for this mark.
32
−
Taking limit as 𝑘 → ∞ for their expression of the form a x (so
1
=0oe is B0). Implied by, for example,
3
2 − 32
lk i m→ − k − ... = 0 − ... but not, for example, for
3
2 − 3
− k 2 −...=0−... without clear use of limiting process.
3
cao from correct integrated expression and finite upper limit (so
dependent on both previous M marks but not the B mark). Accept
equivalent exact forms e.g. 12.
27
6 In this question you must show detailed reasoning.\\
Determine the exact value of $\int _ { 9 } ^ { \infty } \frac { 18 } { x ^ { 2 } \sqrt { x } } \mathrm {~d} x$.
\hfill \mbox{\textit{OCR Further Pure Core 1 2024 Q6 [4]}}