Answer :

Let, I =


I =


If we assume x2 to be an another variable, we can simplify the integral as derivative of x2 i.e. x is present in numerator.


Let, x2 = u


2x dx = du


x dx = 1/2 du


I =


As,


Let, u = A(2u + 1) + B


u = 2Au + A + B


On comparing both sides –


We have,


2A = 1 A = 1/2


A + B = 0 B = –A = –1/2


Hence,


I =


I =


Let, I1and I2 =


Now, I = I1 + I2 ….eqn 1


We will solve I1 and I2 individually.


As, I1 =


Let v = u2 + u + 1 dv = (2u + 1)du


I1 reduces to


Hence,


I1 = { }


On substituting value of u, we have:


I1 = ….eqn 2


As, I2 = and we don’t have any derivative of function present in denominator. we will use some special integrals to solve the problem.


As denominator doesn’t have any square root term. So one of the following two integrals will solve the problem.



Now we have to reduce I2 such that it matches with any of above two forms.


We will make to create a complete square so that no individual term of x is seen in denominator.


I2 =


I2 =


Using: a2 + 2ab + b2 = (a + b)2


We have:


I2 =


I2 =


…eqn 3


From eqn 1, we have:


I = I1 + I2


Using eqn 2 and 3, we get –


I =


Putting value of u in I:


I =


I =


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