How can Arctan and ln be used to simplify the integral of (x+1)/(x^2+4x+5)?

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SUMMARY

The integral ∫(x+1)/(x^2 + 4x + 5) dx can be simplified using the techniques of integration involving Arctan and natural logarithm (ln). The solution involves rewriting the integral as 1/2∫(2x+2)/(x^2+4x+5) dx, which can be further decomposed into two parts: 1/2∫(2x+4)/(x^2+4x+5) dx and 1/2∫(-2)/(1+(x+2)^2) dx. The final result is expressed as [1/2 ln |(x^2+4x+5)|] - arctan(x+2) plus the constant of integration.

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∫(x+1)/(x^2 + 4x +5) dx
Anyone can help me doing it using Arctan and ln
 
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Let's first hear out your own idea to solve this problem? What strategy you have in mind?
 
blue_leaf77 said:
Let's first hear out your own idea to solve this problem? What strategy you have in mind?
∫(x+1)/(x^2+4x+5) dx = 1/2∫ (2x+2)/(x^2+4x+5) dx = 1/2∫(2x+4)/(x^2+4x+5) dx + 1/2∫(-2)/(1+(x+2)^2) dx
[1/2 ln l(x^2+4x+5)l ] - arctan(x+2)
SO??
 
Don't forget the "constant of integration".
 
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HallsofIvy said:
Don't forget the "constant of integration".
Thaanks
 
Try re-writing the original expression into something more manageable.
 

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