Integrate -x/(x^2+5): -1/2ln|x^2+5|

[Natasha1]

Could anyone explain to me very simply by means of a mechanical (formula) approach maybe why the integration of -x / (x^2 + 5) gives - 1/2 ln l x^2 + 5 l

 

[VietDao29]

Could anyone explain to me very simply by means of a mechanical (formula) approach maybe why the integration of -x / (x^2 + 5) gives - 1/2 ln l x^2 + 5 l

One big advice for you, Natasha1 is that, you should open your book, and re-read the chapter that teaches you the u-substitution. read and try to understand the concept, then move on to some examples, try to understand them. And finally, you should pratice solving some integrals that involve the u-substitution.
For this problem, you should let u = x² + 5 => du = 2xdx
So the whole integral becomes:
- \int \frac{xdx}{x ^ 2 + 5} = - \frac{1}{2} \int \frac{du}{u}.
Can you go from here?
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But please, hear me, it won't do any harm to you if you try to re-read the textbook, and try to understand it...