integral from 2 to 5 of pi(ln(7x))^2

解答

\int_{2}^{5} π (ln (7 x))^{2} d x

π (- 2 ln^{2} (14) + ln (\frac{16}{1148916015625}) + 5 ln^{2} (35) + 6)

十进制

95.11486 \dots

求解步骤

\int_{2}^{5} π (ln (7 x))^{2} d x

提出常数:

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

= π \cdot \int_{2}^{5} (ln (7 x))^{2} d x

使用分布积分法

= π [x ln^{2} (7 x) - \int 2 ln (7 x) d x]_{2}^{5}

\int 2 ln (7 x) d x = 2 (x ln (7 x) - x)

= π [x ln^{2} (7 x) - 2 (x ln (7 x) - x)]_{2}^{5}

带入上下限计算后得:

- 2 ln^{2} (14) + 4 ln (14) + 5 ln^{2} (35) + 6 - 10 ln (35)

= π (- 2 ln^{2} (14) + 4 ln (14) + 5 ln^{2} (35) + 6 - 10 ln (35))

化简

= π (- 2 ln^{2} (14) + ln (\frac{16}{1148916015625}) + 5 ln^{2} (35) + 6)

\frac{d}{d a} (5 a^{9} b^{3} + 15 a^{6} b^{7})

\int sec^{6} (x) tan^{3} (x) d x

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

\frac{d}{d x} (\frac{7 x}{2 ^{x}})

\frac{d y}{d x} = 8 x^{7} - 2 sec^{2} (x)