integral from 1 to 2 of ((x+2)/(x+7x^2))

解答

\int_{1}^{2} (\frac{x + 2}{x + 7 x ^{2}}) d x

\frac{53 ln ( 2 ) - 13 ln ( 15 )}{7}

十进制

0.21887 \dots

求解步骤

\int_{1}^{2} \frac{x + 2}{x + 7 x ^{2}} d x

乘开

\frac{x + 2}{x + 7 x ^{2}} : \frac{x}{x + 7 x ^{2}} + \frac{2}{x + 7 x ^{2}}

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{1}^{2} \frac{x}{x + 7 x ^{2}} d x + \int_{1}^{2} \frac{2}{x + 7 x ^{2}} d x

\int_{1}^{2} \frac{x}{x + 7 x ^{2}} d x = \frac{1}{7} (ln (15) - 3 ln (2))

\int_{1}^{2} \frac{2}{x + 7 x ^{2}} d x = 2 (4 ln (2) - ln (15))

= \frac{1}{7} (ln (15) - 3 ln (2)) + 2 (4 ln (2) - ln (15))

化简

\frac{1}{7} (ln (15) - 3 ln (2)) + 2 (4 ln (2) - ln (15)) : \frac{53 ln ( 2 ) - 13 ln ( 15 )}{7}

= \frac{53 ln ( 2 ) - 13 ln ( 15 )}{7}

\int_{0}^{\frac{π}{3}} 5 sin (3 x + π) d x

\int_{0}^{1} x^{2} e^{x - 1} d x

\int_{0}^{7} 8 x 3 x + 1 d x

\int_{3}^{5} - x^{2} + 8 x - 12 d x

\int_{2}^{8} \frac{e ^{\frac{1}{x}}}{x ^{2}} d x