integral from 1 to 4 of 7/(xln(7x))

解答

\int_{1}^{4} \frac{7}{x ln ( 7 x )} d x

7 (ln (ln (28)) - ln (ln (7)))

十进制

3.76533 \dots

求解步骤

\int_{1}^{4} \frac{7}{x ln ( 7 x )} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 7 \cdot \int_{1}^{4} \frac{1}{x ln ( 7 x )} d x

使用换元积分法

= 7 \cdot \int_{7}^{28} \frac{1}{u ln ( u )} d u

使用换元积分法

= 7 \cdot \int_{l n (7)}^{l n (28)} \frac{1}{v} d v

使用积分常见定则:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 7 [ln ∣ v ∣]_{l n (7)}^{l n (28)}

带入上下限计算后得:

ln (ln (28)) - ln (ln (7))

= 7 (ln (ln (28)) - ln (ln (7)))

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