Question

{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
b. Expand sin t² and cos t² in a Maclaurin series, and then integrate to find the first four nonzero terms of the Maclaurin series for S and C.

Accepted Answer

Recall the Maclaurin series expansions for sine and cosine functions: $$ \sin z = \sum_{n=0}^\infty (-1)^n \frac{z^{2n+1}}{(2n+1)!} $$ and $$ \cos z = \sum_{n=0}^\infty (-1)^n \frac{z^{2n}}{(2n)!} . $$Substitute $$ z = t^2 $$ into these series to get the expansions for $$ \sin t^2 $$ and $$ \cos t^2 $$:

$$ \sin t^2 = \sum_{n=0}^\infty (-1)^n \frac{t^{4n+2}}{(2n+1)!} \quad 	ext{and} \quad \cos t^2 = \sum_{n=0}^\infty (-1)^n \frac{t^{4n}}{(2n)!} . $$Write out the first four nonzero terms explicitly for both series:

For $$ \sin t^2 $$:
$$ t^2 - \frac{t^6}{3!} + \frac{t^{10}}{5!} - \frac{t^{14}}{7!} + \cdots $$

For $$ \cos t^2 $$:
$$ 1 - \frac{t^4}{2!} + \frac{t^8}{4!} - \frac{t^{12}}{6!} + \cdots . $$Integrate each term of the series for $$ \sin t^2 $$ from 0 to $$ x  to $$find the Maclaurin series for $$ S(x) = \int_0^x \sin t^2 \, dt $$:

$$ \int_0^x t^{m} dt = \frac{x^{m+1}}{m+1} $$, so integrate term-by-term accordingly. Similarly, integrate each term of the series for $$ \cos t^2 $$ from 0 to $$ x  to $$find the Maclaurin series for $$ C(x) = \int_0^x \cos t^2 \, dt $$, again using term-by-term integration and the power rule for integrals.

{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
b. Expand sin t² and cos t² in a Maclaurin series, and then integrate to find the first four nonzero terms of the Maclaurin series for S and C.

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{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integralsS(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dtb. Expand sin t² and cos t² in a Maclaurin series, and then integrate to find the first four nonzero terms of the Maclaurin series for S and C.

Risposta video verificata per un problema simile:

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{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
b. Expand sin t² and cos t² in a Maclaurin series, and then integrate to find the first four nonzero terms of the Maclaurin series for S and C.