Inverse Trig Integration - Can I use u-substitution?

[opticaltempest]

Here is the problem and the solution manual's solution to the problem. I couldn't see that I had to add 6 and subtract 6 in the numerator then split the integral into two separate integrals. Is there an easier way to do it using u-substitution? Are their alternative ways to evaluate this integral?

http://img466.imageshack.us/img466/3753/problem0lw.jpg

Thanks

 

[siddharth]

From my experience, adding and subtracting 6 is the standard and easiest way to proceed.

The clue to add and subtract 6 is that, you want to express the numerator as the complete derivative of the denominator.

ie,
$$\frac{d}{dx} (x^2 + 6x +13) = 2x + 6$$
Which enables you to use the substitution $x^2 + 6x +13 = u$.

The second part, ie, $$-6 \int \frac{1}{x^2 + 6x +13} dx$$ is then a standard integral.