Answer to question 4.1.21 all subparts

Answer to 4.1.21 (a) :

In this question, we need to transform the denominator in the form (x+m)^2 +n as already discussed while answering the question 4.1.15 .
After transformation, we obtain :

Now the Integrand can be integrated using the standard formula

We have u=(x-3) and a = squareroot(4) = 2. Putting these values in the standard formula, we obtain the answer as :

I use the notation inverse tan instead of arc tan.

Answer to 4.1.21 (b) :

 This is just a normal elementary question, we proceed as follows :

Answer to 4.1.21 (c) : 

Like the previous question, we proceed as follows :

Answer to 4.1.21 (d) :

Split the numerator into 2+2x^2 + x^2 and proceed like the previous questions. Some steps are given below :

Now, all you have to do is expand the integrand into two parts, on the first part (1+x^2) will be canceled and on the second part, x^2 will be canceled. Then, integrate the simplified integrand to obtain the answer :