| Date | May 2018 | Marks available | 2 | Reference code | 18M.1.hl.TZ2.6 |
| Level | HL only | Paper | 1 | Time zone | TZ2 |
| Command term | Find | Question number | 6 | Adapted from | N/A |
Question
Consider the functions \(f,\,\,g,\) defined for \(x \in \mathbb{R}\), given by \(f\left( x \right) = {{\text{e}}^{ - x}}\,{\text{sin}}\,x\) and \(g\left( x \right) = {{\text{e}}^{ - x}}\,{\text{cos}}\,x\).
Find \(f'\left( x \right)\).
Find \(g'\left( x \right)\).
Hence, or otherwise, find \(\int\limits_0^\pi {{{\text{e}}^{ - x}}\,{\text{sin}}\,x\,{\text{d}}x} \).
Markscheme
attempt at product rule M1
\(f'\left( x \right) = - {{\text{e}}^{ - x}}\,{\text{sin}}\,x + {{\text{e}}^{ - x}}\,{\text{cos}}\,x\) A1
[2 marks]
\(g'\left( x \right) = - {{\text{e}}^{ - x}}\,{\text{cos}}\,x - {{\text{e}}^{ - x}}\,{\text{sin}}\,x\) A1
[1 mark]
METHOD 1
Attempt to add \(f'\left( x \right)\) and \(g'\left( x \right)\) (M1)
\(f'\left( x \right) + g'\left( x \right) = - 2{{\text{e}}^{ - x}}\,{\text{sin}}\,x\) A1
\(\int\limits_0^\pi {{{\text{e}}^{ - x}}\,{\text{sin}}\,x\,{\text{d}}x} = \left[ { - \frac{{{{\text{e}}^{ - x}}}}{2}\left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)} \right]_0^\pi \) (or equivalent) A1
Note: Condone absence of limits.
\( = \frac{1}{2}\left( {1 + {{\text{e}}^{ - \pi }}} \right)\) A1
METHOD 2
\(I = \int {{{\text{e}}^{ - x}}} \,{\text{sin}}\,x\,{\text{d}}x\)
\( = - {{\text{e}}^{ - x}}\,{\text{cos}}\,x - \int {{{\text{e}}^{ - x}}} \,{\text{cos}}\,x\,{\text{d}}x\) OR \( = - {{\text{e}}^{ - x}}\,{\text{sin}}\,x + \int {{{\text{e}}^{ - x}}} \,{\text{cos}}\,x\,{\text{d}}x\) M1A1
\( = - {{\text{e}}^{ - x}}\,{\text{sin}}\,x - {{\text{e}}^{ - x}}\,{\text{cos}}\,x - \int {{{\text{e}}^{ - x}}} \,{\text{sin}}\,x\,{\text{d}}x\)
\(I = \frac{1}{2}{{\text{e}}^{ - x}}\left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)\) A1
\(\int_0^\pi {{{\text{e}}^{ - x}}\,{\text{sin}}\,x\,{\text{d}}x = \frac{1}{2}\left( {1 + {{\text{e}}^{ - \pi }}} \right)} \) A1
[4 marks]
