\int_(1)^(7) (1)/(x^(8)) d x \int_{1}^{7} \frac{1}{x^{8}} d x​

Question

`\int_(1)^(7) (1)/(x^(8)) d x`
\int_{1}^{7} \frac{1}{x^{8}} d x​

Answer 1

Use the power rule,

`\displaystyle \int x^n \, dx = (x^(n+1))/(n+1) + C`

with
`n=-8`. It follows by the fundamental theorem of calculus that

`\displaystyle \int_1^7 (dx)/(x^8) = -\frac1{7x^7} \bigg|_(x=1)^(x=7) = -\frac17 \left(\frac1{7^7} - \frac1{1^7}\right) = \boxed{\frac17 - \frac1{7^8}}`