高専の数学解説

第9章 9.13

(1)

I1=∫1elog⁡x dx=[xlog⁡x−x]1e=elog⁡e−e−log⁡1+1=1I_1 = \int_1^e \log x \, dx = [x\log x - x]_1^e = e\log e - e - \log 1 + 1 = 1

(2)

In=∫1e(log⁡x)ndx=[x(log⁡x)n]1e−∫1ex⋅n(log⁡x)n−1⋅1xdx=e(log⁡e)n−log⁡1−n∫1e(log⁡x)n−1dx=e−nIn−1I_n = \int_1^e (\log x)^n dx = [x(\log x)^n]_1^e - \int_1^e x \cdot n(\log x)^{n-1} \cdot \frac{1}{x} dx = e(\log e)^n - \log 1 - n\int_1^e (\log x)^{n-1} dx = e - nI_{n-1}

(3)

I4=e−4I3=e−4(e−3I2)=e−4e+12(e−2I1)=−3e+12e−24I1=9e−24I_4 = e - 4I_3 = e - 4(e - 3I_2) = e - 4e + 12(e - 2I_1) = -3e + 12e - 24I_1 = 9e - 24

解説: r31bn1z