First slide

Evaluation of definite integrals

Question

The absolute value of $\int_{10}^{19} \frac{\sin x}{1 + x^{8}} d x$ is

Moderate

A

less than $10^{- 7}$
B

more than $10^{- 7}$
C

equal to $10^{- 7}$
D

none of these

Solution

For $x \geq 10,$ we have $| \sin x | \leq 1$ and $1 + x^{8} \geq 10^{8}$
$\Rightarrow \frac{1}{1 + x^{8}} \leq 10^{- 8}$
Therefor
$\begin{array}{l} 1 \int_{10}^{19} \frac{\sin x}{1 + x^{8}} d x ∣ \leq \int_{10}^{19} \frac{| \sin x |}{1 + x^{8}} d x \leq \int_{10}^{19} 10^{- 8} d x \\ = 9 (10^{- 8}) < 10^{- 7} \end{array}$

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