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Question Number 162240 by Gbenga last updated on 27/Dec/21

$$\int \frac{2\mathit{x}-5}{{\mathit{x}}^2+4\mathit{x}+5}\mathit{d}\mathit{x}$$

Answered by Ar Brandon last updated on 27/Dec/21

$$2x-5=\alpha (2x+4)+\beta =2\alpha x+(4\alpha +\beta )$$$$\Rightarrow \alpha =1,4+\beta =-5,\beta =-9$$$$\int \frac{2x-5}{x^2+4x+5}dx=\int \frac{2x+4}{x^2+4x+5}dx-9\int \frac{dx}{x^2+4x+5}$$$$=log(x^2+4x+5)-9\int \frac{dx}{{(x+2)}^2+1}$$$$=log(x^2+4x+5)-9arctan(x+2)+C$$

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