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