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Q. The absolute value of $\int\limits_{10}^{19} \frac{(\sin x) d x}{\left(1+x^8\right)}$ is less than

Integrals

Solution:

$\left|\int\limits_{10}^{19} \frac{\sin x}{1+x^8}\right| \leq \int\limits_{10}^{19} \frac{|\sin x|}{1+x^8} d x \leq \int\limits_{10}^{19} \frac{d x}{1+x^8}<\int\limits_{10}^{19} \frac{d x}{x^8}=\left[\frac{x^{-7}}{-7}\right]_{10}^{19}$
$=-\frac{1}{7}\left[19^{-7}-10^{-7}\right]=\frac{1}{7}\left[10^{-7}-19^{-7}\right]<10^{-7}$