19M.1.AHL.TZ2.H_4

pestleMathematicsAAHLPaper 119M· ahl-5-16-integration-by-substitution-parts-and-repeated-partssource ↗

Using the substitution $u = {\text{sin}}\,x$ , find $\int {\frac{{{\text{co}}{{\text{s}}^3}x\,{\text{d}}x}}{{\sqrt {{\text{sin}}\,x} }}}$ .

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$u = {\text{sin}}\,x \Rightarrow {\text{d}}u = {\text{cos}}\,x{\text{d}}x$ (A1)

valid attempt to write integral in terms of $u$ and ${\text{d}}u$ M1

$\int {\frac{{{\text{co}}{{\text{s}}^3}x\,{\text{d}}x}}{{\sqrt {{\text{sin}}\,x} }}} = \int {\frac{{\left( {1 - {u^2}} \right){\text{d}}u}}{{\sqrt u }}}$ A1

$= \int {\left( {{u^{ - \frac{1}{2}}} - {u^{\frac{3}{2}}}} \right)} \,{\text{d}}u$

$= 2{u^{\frac{1}{2}}} - \frac{{2{u^{\frac{5}{2}}}}}{5}\left( { + c} \right)$ (A1)

$= 2\sqrt {{\text{sin}}\,x} - \frac{{2{{\left( {\sqrt {{\text{sin}}\,x} } \right)}^5}}}{5}\left( { + c} \right)$ or equivalent A1

[5 marks]

Examiners’ report

[N/A]