Question
Given that \(\int_0^5 {\frac{2}{{2x + 5}}} {\rm{d}}x = \ln k\) , find the value of k .
Answer/Explanation
Markscheme
correct integration, \(2 \times \frac{1}{2}\ln (2x + 5)\) A1A1
Note: Award A1 for \(2 \times \frac{1}{2}( = 1)\) and A1 for \(\ln (2x + 5)\) .
evidence of substituting limits into integrated function and subtracting (M1)
e.g. \(\ln (2 \times 5 + 5) – \ln (2 \times 0 + 5)\)
correct substitution A1
e.g. \(\ln 15 – \ln 5\)
correct working (A1)
e.g. \(\ln \frac{{15}}{5},\ln 3\)
\(k = 3\) A1 N3
[6 marks]
Question
Let \(f(x) = \int {\frac{{12}}{{2x – 5}}} {\rm{d}}x\) , \(x > \frac{5}{2}\) . The graph of \(f\) passes through (\(4\), \(0\)) .
Find \(f(x)\) .
Answer/Explanation
Markscheme
attempt to integrate which involves \(\ln \) (M1)
eg \(\ln (2x – 5)\) , \(12\ln 2x – 5\) , \(\ln 2x\)
correct expression (accept absence of \(C\))
eg \(12\ln (2x – 5)\frac{1}{2} + C\) , \(6\ln (2x – 5)\) A2
attempt to substitute (4,0) into their integrated f (M1)
eg \(0 = 6\ln (2 \times 4 – 5)\) , \(0 = 6\ln (8 – 5) + C\)
\(C = – 6\ln 3\) (A1)
\(f(x) = 6\ln (2x – 5) – 6\ln 3\) \(\left( { = 6\ln \left( {\frac{{2x – 5}}{3}} \right)} \right)\) (accept \(6\ln (2x – 5) – \ln {3^6}\) ) A1 N5
Note: Exception to the FT rule. Allow full FT on incorrect integration which must involve \(\ln\).
[6 marks]
Question
Let \(f\left( x \right) = 6{x^2} – 3x\). The graph of \(f\) is shown in the following diagram.
Find \(\int {\left( {6{x^2} – 3x} \right){\text{d}}x} \).
Find the area of the region enclosed by the graph of \(f\), the x-axis and the lines x = 1 and x = 2 .
Answer/Explanation
Markscheme
\(2{x^3} – \frac{{3{x^2}}}{2} + c\,\,\,\left( {{\text{accept}}\,\,\frac{{6{x^3}}}{3} – \frac{{3{x^2}}}{2} + c} \right)\) A1A1 N2
Notes: Award A1A0 for both correct terms if +c is omitted.
Award A1A0 for one correct term eg \(2{x^3} + c\).
Award A1A0 if both terms are correct, but candidate attempts further working to solve for c.
[2 marks]
substitution of limits or function (A1)
eg \(\int_1^2 {f\left( x \right)} \,{\text{d}}x,\,\,\left[ {2{x^3} – \frac{{3{x^2}}}{2}} \right]_1^2\)
substituting limits into their integrated function and subtracting (M1)
eg \(\frac{{6 \times {2^3}}}{3} – \frac{{3 \times {2^2}}}{2} – \left( {\frac{{6 \times {1^3}}}{3} + \frac{{3 \times {1^2}}}{2}} \right)\)
Note: Award M0 if substituted into original function.
correct working (A1)
eg \(\frac{{6 \times 8}}{3} – \frac{{3 \times 4}}{2} – \frac{{6 \times 1}}{3} + \frac{{3 \times 1}}{2},\,\,\left( {16 – 6} \right) – \left( {2 – \frac{3}{2}} \right)\)
\(\frac{{19}}{2}\) A1 N3
[4 marks]