Answer to question 4.1.15

There is a trick for doing this type of question having denominator of integrand in the form ax^2 + bx + c :

We transform the  denominator in the form (x + m)^2 + n and the first thing you have to do is decide m and this is how you do it : select m in such a way that when you expand (x+m)^2, the coefficient of x will be equal to the coefficient of x in the equation ax^2 + bx + c and then select n so that the constant term while expanding (x+m)^2 is equal to c  in the equation.

Here's how we transform the integrand using the above method :

On the above step, notice how we chose 1/2 and 3/4 as m and n. Now, the integrand can be transformed to use the standard integration formula

This is how we transform our integrand :

Now, let's use the standard formula. We have our u=(x+1/2) and a =square_root(3)/2 . This gives us :

 

Now, after simplification, we obtain the answer :