Evaluating the integral of 1 divided by x (x^2+4)^2 dx

Question

Evaluating the integral of 1 divided by x (x^2+4)^2 dx

Answer 1

Dear Robert,

When you have an integral of this form where the denominator is a product of functions, it is advisable to decompose into partial fractions if applicable.

 

Set it up in the form

1/ [x (x^2+4)^2 ] = A/ x + (Bx +C)/ (x^2+4) + (Dx + E)/ (x^2+4)^2

 

Solve for A, B, C , D, and E.

 

You should obtain the following:

A = 1/16

B = - 1/16

C = 0

D = -1/4

E = 0

 

You now have the integral of a single complex function as the sum of the integrals of three simpler functions.

The first two are of ln form; the last one is a power function.

I leave this as an exercise for you to do.

 

Please note, I am available for online tutoring. If you need help in decomposing a function into partial fractions, or general integration, or most of Calculus,  contact me through Wyzant .

 

Best Wishes!

Hilton Tottaram

Answer 2

Dear Robert:

 

I don't do another person's homework, but to help you learn the material I will give you some hints (I left out some constants so you must work them out).

 

In general, when there is a term (x^2 + or - a^2) one thing to try is trig substitution. In this case you have the term

(x^2 + 4) which can be written as (x^2 + 2^2). So try trig substitution. The plus sign means that sqrt(x^2 + 2^2) should be the hypotenuse of a right triangle. Then choose the side x to be the opposite side to the angle a. That means the adjacent side to the angle a must be 2. So you have x/2 = tan a. Change the integration variable from x to a. When you do that you should get an integral of the form (I left out a constant)

 

[(cos^3 a)/(sin a)] da

 

Now change of variable again y = sin a.

 

Good luck