Standard +0.8 This requires executing a non-trivial substitution with trigonometric identities (sin 2x = 2sin x cos x), careful handling of the derivative and limits, and algebraic manipulation to reach the exact form a + b√2. While systematic, it demands multiple coordinated steps beyond routine integration practice.
8 Use the substitution \(\mathrm { u } = 1 - \sin \mathrm { x }\) to find the exact value of
$$\int _ { \pi } ^ { \frac { 3 } { 2 } \pi } \frac { \sin 2 x } { \sqrt { 1 - \sin x } } d x$$
Give your answer in the form \(\mathrm { a } + \mathrm { b } \sqrt { 2 }\) where \(a\) and \(b\) are rational numbers to be determined.
Correctly use limits \(u=2\) and \(0\) in expression of form \(au^{\frac{1}{2}} + bu^{\frac{3}{2}}\), OR limits \(x = \frac{3}{2}\pi\) and \(\frac{1}{2}\pi\) in expression of form \(a(1-\sin x)^{\frac{1}{2}} + b(1-\sin x)^{\frac{3}{2}}\)
DM1
Obtain \(\dfrac{8}{3} - \dfrac{4}{3}\sqrt{2}\)
A1
## Question 8:
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $du = -\cos x\, dx$ | B1 | |
| Use $\sin 2x = 2\sin x\cos x$ and write integral in terms of $u$ | \*M1 | |
| Obtain $\pm 2\displaystyle\int \dfrac{(1-u)}{\sqrt{u}}\, du$ or equivalent | A1 | |
| Integrate correctly to obtain $au^{\frac{1}{2}} + bu^{\frac{3}{2}}$ | DM1 | |
| Obtain correct $-4u^{\frac{1}{2}} + \dfrac{4}{3}u^{\frac{3}{2}}$ | A1 | |
| Correctly use limits $u=2$ and $0$ in expression of form $au^{\frac{1}{2}} + bu^{\frac{3}{2}}$, OR limits $x = \frac{3}{2}\pi$ and $\frac{1}{2}\pi$ in expression of form $a(1-\sin x)^{\frac{1}{2}} + b(1-\sin x)^{\frac{3}{2}}$ | DM1 | |
| Obtain $\dfrac{8}{3} - \dfrac{4}{3}\sqrt{2}$ | A1 | |
8 Use the substitution $\mathrm { u } = 1 - \sin \mathrm { x }$ to find the exact value of
$$\int _ { \pi } ^ { \frac { 3 } { 2 } \pi } \frac { \sin 2 x } { \sqrt { 1 - \sin x } } d x$$
Give your answer in the form $\mathrm { a } + \mathrm { b } \sqrt { 2 }$ where $a$ and $b$ are rational numbers to be determined.\\
\hfill \mbox{\textit{CAIE P3 2024 Q8 [7]}}