Edexcel AEA 2017 June — Question 6 16 marks

Exam BoardEdexcel
ModuleAEA (Advanced Extension Award)
Year2017
SessionJune
Marks16
PaperDownload PDF ↗
TopicIntegration using inverse trig and hyperbolic functions
TypeDerivative of inverse hyperbolic function
DifficultyChallenging +1.8 This is an AEA question requiring multiple advanced techniques: proving derivative of inverse hyperbolic function, non-standard substitution with inverse trig integration, partial fractions, and combining all methods for a definite integral. While systematic, it demands strong technical facility across several Further Maths topics and careful algebraic manipulation through multiple steps.
Spec1.02y Partial fractions: decompose rational functions1.08h Integration by substitution1.08i Integration by parts1.08j Integration using partial fractions
6.(a)Show that $$\frac { \mathrm { d } } { \mathrm {~d} u } \ln \left( u + \sqrt { u ^ { 2 } - 1 } \right) = \frac { 1 } { \sqrt { u ^ { 2 } - 1 } }$$ (b)Use the result from part(a)and the substitution \(x + 3 = \frac { 1 } { t }\) to find $$\int \frac { 1 } { ( x + 3 ) \sqrt { 2 x + 7 } } \mathrm {~d} x$$ (6)
(c)Express \(\frac { 1 } { 2 x ^ { 2 } + 13 x + 21 }\) in partial fractions.
(d)Find $$\int _ { 1 } ^ { 9 } \frac { 1 } { \left( 2 x ^ { 2 } + 13 x + 21 \right) \sqrt { 2 x + 7 } } \mathrm {~d} x$$ giving your answer in the form \(\ln r - s\) where \(r\) and \(s\) are rational numbers.
6.(a)Show that

$$\frac { \mathrm { d } } { \mathrm {~d} u } \ln \left( u + \sqrt { u ^ { 2 } - 1 } \right) = \frac { 1 } { \sqrt { u ^ { 2 } - 1 } }$$

(b)Use the result from part(a)and the substitution $x + 3 = \frac { 1 } { t }$ to find

$$\int \frac { 1 } { ( x + 3 ) \sqrt { 2 x + 7 } } \mathrm {~d} x$$

(6)\\
(c)Express $\frac { 1 } { 2 x ^ { 2 } + 13 x + 21 }$ in partial fractions.\\
(d)Find

$$\int _ { 1 } ^ { 9 } \frac { 1 } { \left( 2 x ^ { 2 } + 13 x + 21 \right) \sqrt { 2 x + 7 } } \mathrm {~d} x$$

giving your answer in the form $\ln r - s$ where $r$ and $s$ are rational numbers.\\

\hfill \mbox{\textit{Edexcel AEA 2017 Q6 [16]}}
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Q1 7 Q2 9 Q3 13 Q4 13 Q5 14 Q6 16 Q7 21