solve-x-1-x-2-7x-3-dx-

Question Number 161101 by bounhome last updated on 12/Dec/21

s o l v e :

\int \frac{x + 1}{x^{2} - 7 x - 3} d x

Answered by FongXD last updated on 12/Dec/21

= \frac{1}{2} \int \frac{2 x - 7 + 9}{x^{2} - 7 x - 3} dx

= \frac{1}{2} \int \frac{2 x - 7}{x^{2} - 7 x - 3} dx + \frac{9}{2} \int \frac{dx}{x^{2} - 7 x - 3}

= \frac{1}{2} \ln ∣ x^{2} - 7 x - 3 ∣ + \frac{9}{2} \int \frac{dx}{{(x - \frac{7}{2})}^{2} - \frac{61}{4}}

= \frac{1}{2} \ln ∣ x^{2} - 7 x - 3 ∣ + \frac{9}{2} \int \frac{dx}{(x - \frac{7}{2} - \frac{\sqrt{61}}{2}) (x - \frac{7}{2} + \frac{\sqrt{61}}{2})}

= \frac{1}{2} \ln ∣ x^{2} - 7 x - 3 ∣ + \frac{9}{2 \sqrt{61}} \int (\frac{1}{x - \frac{7}{2} - \frac{\sqrt{61}}{2}} - \frac{1}{x - \frac{7}{2} + \frac{\sqrt{61}}{2}}) dx

= \frac{1}{2} \ln ∣ x^{2} - 7 x - 3 ∣ + \frac{9 \sqrt{61}}{122} \ln ∣ \frac{2 x - 7 - \sqrt{61}}{2 x - 7 + \sqrt{61}} ∣ + c, c \in R

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