Answer to Question 4.1.18

The denominator of this question needs to be transformed in the form (x+m)^2 + n (this form will be inside the square root) (please see question 4.1.15 here to know how you can find the values of m and n) and then we can use the standard integration formula :

After choosing values of m and n, the transformed equation looks like this :

Now we can use the standard integration formula. We have u=(x+3) and a = squareroot(8). This yields the answer :

NOTE : I use log instead of ln most of the time. So when I use log, consider it as ln(i.e natural log of base e).

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 :

Answer to question 4.1.14

Transforming the integrand into the standard form 1/(u^2 + a^2) will enable us to use the standard integration formula :

The transformation is carried out as follows :

Taking 4 common from the denominator to make 1 as the coefficient of x^2 (since coefficient or x^2 is one in the standard formula), we get :

Now, let's use the standard formula to integrate. We have u=x and a = (5/2)

Answer to question 4.1.7

Multiplying by conjugate on both numerator and denominator yields the result as shown below.

Now, separately integrate each term of the simplified integrand. Doing this, we get :

Answer to question 4.1.2

This is a simple problem. Use long division and integrate the simplified form. See the solution below :

, We get

 Divisor(D) = 2x - 1 ;Remainder(R) = 1 and Quotient(Q) = 3x^2 +2x

Now I becomes :

On the above step what I did is replaced the polynomial by Q+R/D. Now you can Integrate using direct formulas :

Note : I use log instead of ln( log to base e).