18M.1.AHL.TZ2.H_6

pestleMathematicsAIHLPaper 118M· ahl-5-11-indefinite-integration-reverse-chain-by-substitutionsource ↗

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 $f'\left( x \right)$ .

[2]

a.i.

Find $g'\left( x \right)$ .

[1]

a.ii.

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

[4]

Markscheme / solution

attempt at product rule M1

$f'\left( x \right) = - {{\text{e}}^{ - x}}\,{\text{sin}}\,x + {{\text{e}}^{ - x}}\,{\text{cos}}\,x$ A1

[2 marks]

a.i.

$g'\left( x \right) = - {{\text{e}}^{ - x}}\,{\text{cos}}\,x - {{\text{e}}^{ - x}}\,{\text{sin}}\,x$ A1

[1 mark]

a.ii.

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]

Examiners’ report

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a.i.

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a.ii.

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