Find $f'\left( x \right)$.

Find $g'\left( x \right)$.

Hence, or otherwise, find $\int\limits_0^\pi  {{{\text{e}}^{ - x}}\,{\text{sin}}\,x\,{\text{d}}x}$.

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]