integral from 1 to e of ln(5x^3)

解答

\int_{1}^{e} ln (5 x^{3}) d x

e ln (5) - ln (5) + 3

十进制

5.76546 \dots

求解步骤

\int_{1}^{e} ln (5 x^{3}) d x

使用分布积分法

= [x ln (5 x^{3}) - \int 3 d x]_{1}^{e}

\int 3 d x = 3 x

= [x ln (5 x^{3}) - 3 x]_{1}^{e}

化简

ln (5 x^{3}) : 3 ln (x) + ln (5)

= [x (ln (5) + 3 ln (x)) - 3 x]_{1}^{e}

带入上下限计算后得:

e ln (5) - ln (5) + 3

= e ln (5) - ln (5) + 3