WJEC Unit 3 2019 June — Question 14

Exam BoardWJEC
ModuleUnit 3 (Unit 3)
Year2019
SessionJune
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TopicIntegration by Substitution
TypeSubstitution u = expression involving trig (non-pure sin/cos)
DifficultyModerate -0.3 This is a multi-part integration question covering standard techniques. Parts (a) and (b) are routine applications of basic integration and chain rule recognition. Part (c) requires integration by parts, which is standard. Part (d) provides the substitution explicitly, making it mechanical to execute. While part (d) requires careful handling of limits and the substitution, the overall question demands only textbook techniques with no novel insight, making it slightly easier than average.
Spec1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08h Integration by substitution1.08i Integration by parts
a) Find \(\int \left( \mathrm { e } ^ { 2 x } + 6 \sin 3 x \right) \mathrm { d } x\). b) Find \(\int 7 \left( x ^ { 2 } + \sin x \right) ^ { 6 } ( 2 x + \cos x ) \mathrm { d } x\).
c) Find \(\int \frac { 1 } { x ^ { 2 } } \ln x \mathrm {~d} x\).
d) Use the substitution \(u = 2 \cos x + 1\) to evaluate $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { \sin x } { ( 2 \cos x + 1 ) ^ { 2 } } d x$$ 
a) Find $\int \left( \mathrm { e } ^ { 2 x } + 6 \sin 3 x \right) \mathrm { d } x$.

b) Find $\int 7 \left( x ^ { 2 } + \sin x \right) ^ { 6 } ( 2 x + \cos x ) \mathrm { d } x$.\\
c) Find $\int \frac { 1 } { x ^ { 2 } } \ln x \mathrm {~d} x$.\\
d) Use the substitution $u = 2 \cos x + 1$ to evaluate

$$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { \sin x } { ( 2 \cos x + 1 ) ^ { 2 } } d x$$

\hfill \mbox{\textit{WJEC Unit 3 2019 Q14}}
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