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Problem 72 Hard Difficulty

The sine integral function $$ \displaystyle \text{Si}(x) = \int^x_0 \frac{\sin t}{t} \, dt $$ is important in electrical engineering. [The integrand $ f(t) = (\sin t)/t $ is not defined when $ t = 0 $, but we know that its limit is 1 when $ t \to 0 $. So we defined $ f(0) = 1 $ and this makes $ f $ a continuous function everywhere.]

(a) Draw the graph of $ \text{Si} $.
(b) At what values of $ x $ does this function have local maximum values?
(c) Find the coordinates of the first inflection point to the right of the origin.
(d) Does this function have horizontal asymptotes?
(e) Solve the following equation correct to one decimal place:
$$ \displaystyle \int^x_0 \frac{\sin t}{t} \, dt = 1 $$

Answer

a.
b. $\frac{x \cos x-\sin x}{x^{2}}$
c. $x=k \pi$
d. $\tan x=x$
e. $\pi / 2$

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Video Transcript

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