Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2017 | Oct-Nov | (P2-9709/23) | Q#4

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Question

     i.       Find

   ii.       Given that

find the value of the positive constant a.

Solution

     i.
 

We are required to find;

We have the trigonometric identity;

It can be rearranged as;

Therefore;

  provided that

It can be rearranged as;

Therefore;

Rule for integration of  is:

Rule for integration of  is:

Rule for integration of  is:

 

   ii.
 

We are given that;

Rule for integration of  is:

Since ;

Power Rule;

 and are inverse functions. The composite function is an identity function, with 

domain the positive real numbers. Therefore; 

Taking anti-logarithm of both sides;

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