`int (x^2 - 5x + 16)/((2x + 1)(x - 2)^2) dx` Evaluate the integral

Expert Answers
gsarora17 eNotes educator| Certified Educator


Let's first express the integrand as sum of proper rational expressions by applying partial fraction decomposition,






Now equate the coefficients of the polynomial in the numerator on the both sides,

`A+2B=1` --------------------------------(1)

`-4A-3B+2C=-5`  -----------------(2)

`4A-2B+C=16`  ----------------------(3)

Now let's solve the above three equations by the method of substitution,

From equation 1 :`A=1-2B`

Substitute the above value of A in equation 2 ,




`5B+2C=-1`     ------------------------ (4)

Now substitute the value of A in equation 3,





`-10B+C=12`    -----------------------(5)

Now solve the equations 4 and 5 by the method of elimination,

Multiply equation 4 by 2,

`10B+4C=-2`  ----------------------(6)

Now add the equations 5 and 6,



Plug the value of C in equation 5.




Plug the value of B in equation 1.






Now let's evaluate the above three integrals,


Let's apply the integral substitution:`u=2x+1`




Substitute back u=2x+1,


Now let's evaluate `int1/(x-2)dx`

apply integral substitution: `v=x-2`




substitute back v=x-2,


Now let's evaluate integral `int1/(x-2)^2dx`

apply the integral substitution: t=x-2






Substitute back t=x-2,



`=3/2ln|2x+1|-ln|x-2|-2/(x-2)+C`   where C is a constant


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