Question

$\int_{1}^{7} \frac{1}{x^{8}} d x$
\int_{1}^{7} \frac{1}{x^{8}} d x​

Answer 1

Use the power rule,

$\displaystyle \int x^n \, dx = \frac{x^{n+1}}{n+1} + C$

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

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