2x-5-x-2-4x-5-dx-

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

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

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

2 x - 5 = α (2 x + 4) + β = 2 α x + (4 α + β)

\Rightarrow α = 1, 4 + β = - 5, β = - 9

\int \frac{2 x - 5}{x^{2} + 4 x + 5} d x = \int \frac{2 x + 4}{x^{2} + 4 x + 5} d x - 9 \int \frac{d x}{x^{2} + 4 x + 5}

= l o g (x^{2} + 4 x + 5) - 9 \int \frac{d x}{{(x + 2)}^{2} + 1}

= l o g (x^{2} + 4 x + 5) - 9 a r c t a n (x + 2) + C