integral of (2x+7)/(x^2+7x+2)

해법

\int \frac{2 x + 7}{x ^{2} + 7 x + 2} d x

- \frac{1}{41} (2 x + 7) (ln \frac{1}{41} (2 x + 7) + 1 - ln \frac{1}{41} (2 x + 7) - 1) + \frac{2}{41} x ln \frac{1}{41} (2 x + 7) + 1 - \frac{2}{41} x ln \frac{1}{41} (2 x + 7) - 1 + \frac{7}{41} ln \frac{1}{41} (2 x + 7) + 1 - \frac{7}{41} ln \frac{1}{41} (2 x + 7) - 1 + ln (2 x + 7 + 41 2 x + 7 - 41) + C

솔루션 단계

\int \frac{2 x + 7}{x ^{2} + 7 x + 2} d x

부품별 통합 적용

= - \frac{1}{41} (2 x + 7) (ln \frac{1}{41} (2 x + 7) + 1 - ln \frac{1}{41} (2 x + 7) - 1) - \int - \frac{2}{41} (ln \frac{1}{41} (2 x + 7) + 1 - ln \frac{1}{41} (2 x + 7) - 1) d x

\int - \frac{2}{41} (ln \frac{1}{41} (2 x + 7) + 1 - ln \frac{1}{41} (2 x + 7) - 1) d x = - \frac{2}{41} x ln \frac{1}{41} (2 x + 7) + 1 + \frac{2}{41} x ln \frac{1}{41} (2 x + 7) - 1 - \frac{7}{41} ln \frac{1}{41} (2 x + 7) + 1 + \frac{7}{41} ln \frac{1}{41} (2 x + 7) - 1 - ln (2 x + 7 + 41 2 x + 7 - 41)

= - \frac{1}{41} (2 x + 7) (ln \frac{1}{41} (2 x + 7) + 1 - ln \frac{1}{41} (2 x + 7) - 1) - (- \frac{2}{41} x ln \frac{1}{41} (2 x + 7) + 1 + \frac{2}{41} x ln \frac{1}{41} (2 x + 7) - 1 - \frac{7}{41} ln \frac{1}{41} (2 x + 7) + 1 + \frac{7}{41} ln \frac{1}{41} (2 x + 7) - 1 - ln (2 x + 7 + 41 2 x + 7 - 41))

- \frac{1}{41} (2 x + 7) (ln \frac{1}{41} (2 x + 7) + 1 - ln \frac{1}{41} (2 x + 7) - 1) - (- \frac{2}{41} x ln \frac{1}{41} (2 x + 7) + 1 + \frac{2}{41} x ln \frac{1}{41} (2 x + 7) - 1 - \frac{7}{41} ln \frac{1}{41} (2 x + 7) + 1 + \frac{7}{41} ln \frac{1}{41} (2 x + 7) - 1 - ln (2 x + 7 + 41 2 x + 7 - 41))

간소화하다 :

- \frac{1}{41} (2 x + 7) (ln \frac{1}{41} (2 x + 7) + 1 - ln \frac{1}{41} (2 x + 7) - 1) + \frac{2}{41} x ln \frac{1}{41} (2 x + 7) + 1 - \frac{2}{41} x ln \frac{1}{41} (2 x + 7) - 1 + \frac{7}{41} ln \frac{1}{41} (2 x + 7) + 1 - \frac{7}{41} ln \frac{1}{41} (2 x + 7) - 1 + ln (2 x + 7 + 41 2 x + 7 - 41)

= - \frac{1}{41} (2 x + 7) (ln \frac{1}{41} (2 x + 7) + 1 - ln \frac{1}{41} (2 x + 7) - 1) + \frac{2}{41} x ln \frac{1}{41} (2 x + 7) + 1 - \frac{2}{41} x ln \frac{1}{41} (2 x + 7) - 1 + \frac{7}{41} ln \frac{1}{41} (2 x + 7) + 1 - \frac{7}{41} ln \frac{1}{41} (2 x + 7) - 1 + ln (2 x + 7 + 41 2 x + 7 - 41)

솔루션에 상수 추가

= - \frac{1}{41} (2 x + 7) (ln \frac{1}{41} (2 x + 7) + 1 - ln \frac{1}{41} (2 x + 7) - 1) + \frac{2}{41} x ln \frac{1}{41} (2 x + 7) + 1 - \frac{2}{41} x ln \frac{1}{41} (2 x + 7) - 1 + \frac{7}{41} ln \frac{1}{41} (2 x + 7) + 1 - \frac{7}{41} ln \frac{1}{41} (2 x + 7) - 1 + ln (2 x + 7 + 41 2 x + 7 - 41) + C