Question 8 - A-Level Maths

Exam Board	OCR
Module	FP2 (Further Pure Mathematics 2)
Session	Specimen
Topic	Integration with Partial Fractions

Use the substitution $t = \tan \frac { 1 } { 2 } x$ to show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { \frac { 1 - \cos x } { 1 + \sin x } } \mathrm {~d} x = 2 \sqrt { } 2 \int _ { 0 } ^ { 1 } \frac { t } { ( 1 + t ) \left( 1 + t ^ { 2 } \right) } \mathrm { d } t$$
Express $\frac { t } { ( 1 + t ) \left( 1 + t ^ { 2 } \right) }$ in partial fractions.
Hence find $\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { \frac { 1 - \cos x } { 1 + \sin x } } \mathrm {~d} x$, expressing your answer in an exact form.