Fun problem: ? x / (x^2 + 6x + 10) dx

[Matt Jacques]

Fun problem: ? x / (x^2 + 6x + 10) dx

Integration by parts proves 1=1! My mathematical fame is at hand! So how would you do this one?

 

[suyver]

$$\int\frac{x}{x^2 + 6x + 10}{\rm d}x = \frac12\left(\ln[10 + x(6 + x)]-6 \arctan [3 + x] \right)$$


So, what does this have to do with 1=1 (which is selfevidently true anyway)?

 

[HallsofIvy]

Integration by parts proves 1=1? In other words, you used integration by parts twice, the second time reversing your choice for u and dv so the two cancelled!

"Partial fractions" is what you need here. The denominator, x^2 + 6x + 10, is "irreducible" over the real numbers. It is the same as
x^2+ 6x+ 9+ 1= (x+3)^2+ 1. I would recommend the substitution
u= x+ 3 so that du= dx, x= u- 3 and the problem becomes integrating
(u-3)/(u^2+1)= u/(u^2+1)- 3/(u^2+1).

The first of those can be done by the further substitution v= u^2+1 and the second is a simple arctangent.

 

[Matt Jacques]

OOops, I forgot that x could be expressed in terms of u.