Question

i.       Prove that  .

   ii.       Hence

a.  Find the exact value of

b.  Evaluate

Solution

     i.
 

We are given that;

except where  or undefined

  provided that

We have the trigonometric identity;

Therefore;

Hence;

 

   ii.
 

a.
 

We are required to find the exact value of;

As demonstrated in (i);

We can write;

Therefore

b.
 

We are required to evaluate;

As demonstrated in (i);

Therefore;

Rule for integration of  is: