SPS SPS FM 2020 June — Question 2 4 marks

Exam BoardSPS
ModuleSPS FM (SPS FM)
Year2020
SessionJune
Marks4
TopicIntegration by Substitution
TypeTrigonometric substitution: show transformation then evaluate
DifficultyStandard +0.3 This is a straightforward integration by substitution question with explicit guidance. Students are told exactly which substitution to use (x = sin θ), must verify the transformation algebraically using standard trigonometric identities (1 - sin²θ = cos²θ), then integrate sec²θ which is a standard result. While it requires careful algebraic manipulation and knowledge of trigonometric identities, the question provides the roadmap and involves only routine techniques with no problem-solving insight required.
Spec1.05a Sine, cosine, tangent: definitions for all arguments1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.08h Integration by substitution
2. Show that the substitution \(x = \sin \theta\) transforms $$\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x$$ to $$\int \sec ^ { 2 } \theta d \theta$$ and hence find $$\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x$$ 
2. Show that the substitution $x = \sin \theta$ transforms

$$\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x$$

to

$$\int \sec ^ { 2 } \theta d \theta$$

and hence find

$$\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x$$

\hfill \mbox{\textit{SPS SPS FM 2020 Q2 [4]}}
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