Question

Let $I={\int }_{10}^{19}\frac{\sin x}{1+x^8}dx$. Then

• A. $|I|<{10}^{-9}$
• B. $|I|<{10}^{-7}$
• C. $|I|<{10}^{-5}$
• D. $|I|<{10}^{-8}$
  

Answer 1

Answer: (B)

$|I|<{10}^{-7}$

$I={\int }_{10}^{19}\frac{\sin x}{1+x^8}dx$

$|I|=\left|{\int }_{10}^{19}\frac{\sin x}{1+x^8}dx\right|$

$\le {\int }_{10}^{19}\left|\frac{\sin x}{1+x^8}\right|dx$

$\le {\int }_{10}^{19}\frac{1}{|1+x^8|}dx$ ..... $(\because |\sin x|\le 1)$

$<{\int }_{10}^{19}{10}^{-8}dx$ ........ $(\because 1+x^8>{10}^8$ for $10\le x\le 19)$

$=9\times {10}^{-8}<10\times {10}^{-8}={10}^{-7}$

Hence, $|I|<{10}^{-7}$.