Evaluate the following indefinite integral

Question

∫x⁴/(x-3)(x+4) dx

Answer 1

Hi Peter,

 

To do this, you must use "Partial Fraction."  But before all that, you must divide x⁴ and (x+4)(x-3).  In other words, use long division: x⁴+0x³+0x²+0x+0 / (x²+x-12).  This will give you x²-x-11 - [x/(x²+x-12)] - [132/(x²+x-12)] or -132[1/(x²+x-12)].

 

First part (x²+x-12) is fine (integrate as normal) but second and third part is where you have to use "Partial Fraction."

 

First part will be (x³/3) - (x²/2) -11x

 

So, you have x/(x2+x-12)   or   x/(x+4)(x-3).

 

   A   +    B      = x          A(x-3) + B(x+4)   =    x        

(x+4)    (x-3)                     (x+4) (x-3)

 

 

Ax -3A + Bx + 4B = x        Ax+Bx -3A+4B = x   x(A+B) -3A+4B = x

 

From this equation (x(A+B) -3A+4B = x), you can obtain that A+B = -1 & -3A+4B = 0.  From here, you can set these two equations and solve for each.  You can multiply 3 to first equation to cancel A which it will give you 7B = -3.  And that will give you that B = -3/7.  Since A+B = 1, A will become -4/7.  Put these two values into the equation and integrate.  That will give you 

 

(-4/7) ln (x+4) + (-3/7) ln (x-3)      (this is only second part)

 

 

The third part is very similar but with -132:

 

  A + B    = -132     A(x-3) + B(x+4) = -132
(x+4) (x-3)                 (x+4) (x-3)

Ax -3A + Bx + 4B = 1          Ax+Bx -3A+4B = 1            x(A+B) -3A+4B = 1

 

For third part, there is no x.  Therefore, A+B = 0 and -3A+4B = 1.  From here, You can multiply 3 to first equation to cancel A which it will give you 7B = -132.  And that will give you that B= 1/7 and A = -1/7.

 

So, if you integrate third equation -132[1/(x²+x-12)], it will look like this:

 

∫(-132) dx / [(x+4) (x-3)] 

-132∫ dx / [(x+4)(x-3)]     =  -132 { (-1/7)∫ [dx/(x+4)]  + (1/7)∫ [dx/(x-3)] }

 

-132  [-1/7 ln (x+4) + 1/7 ln (x-3)]

 

After combining all 3, you will get:

 

x³/3 - x²/2 - 11x - (4/7) ln (x+4) - (3/7) ln (x-3) -132 [-1/7 ln (x+4) + 1/7 ln (x-3)] + C

 

I hope you got it.  Good luck to you.

 

Yohan Chu   (LV  NV)

 

To check it, take the derivative.  It might take a while.