Estimate the absolute value of the integral `int_(10)^(19)(sinx)/(1+x^8)dx`

Explanation

Since `|sinx|le1,` then
`|(sinx)/(1+x^(8))|le1/(|1+x^(8)|)`………………1
But `10lexle19`. So `1+x^(8)gtx^(8)ge10^(8)`
or `1/(1+x^(8))lt1/(x^(8))le1/(10^(8))`
or `1/(|1+x^(8)|)le1/(10^(8))`…………..2
From 1 and 2 we get `|(sinx)/(1+x^(8))|le10^(-8)`
Then `int_(10)^(19)(sinx)/(1+x^(8))dx le(19-10)xx10^(-8)`
`=9xx10^(-8)=(10-1)xx10^(8)`
`=10^(-7)-10^(-8)lt10^(-7)`
Hence `|int_(10)^(19)(sinxdx)/(1+x^(8))|lt10^(-7)`.