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The Fresnel function $ S(x) = \displaystyle \int_0^x \sin \left(\frac{1}{2} \pi t^2 \right) dt $ was discussed in Example 5.3.3 and is used extensively in the theory of optics. Find $ S(x) dx $. [Your answer will involve $ S(x) $.]

$x S(x)+\frac{1}{\pi} \cos \left(\frac{1}{2} \pi x^{2}\right)+C$

Integration Techniques

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So we want to use integration by parts and that tells us that the integral of f times G prime is equal to F times G um, minus sorry. The f times G minus the integral of g times f prime. So what we have here is SFX equal to the integral from zero to x of sign one half hi t squared the 18th. Um so when we look at what our function is, we see that f of t equals sign of one half pi t square. So when we take the integral of this when we take the derivative of this so we have f prime of X, we see that sets f of X. So s prime of X in this case is just going to give us a fax which again equals justice right here. Um then based on that, we see that s prime of X, which equals f of X, is just going to give us a prime. So now we need our G and G prime. Um, we see that we can just let g FX equal acts. So that way our G prime of X just equals one. So now what we end up having is the integral of SFX times one DX That's the same thing as SFX Times axe. My name is the integral of X times Sign one half pi X squared D x um And then what we do now is we can evaluate this and we end up getting we want to evaluate our integral, so we'll use u substitution. And when we solve this out, what we end up getting is a negative. Um, co sign pi ax squared over two over pie plus c um, so because this was going to be a negative, integral, it's gonna become positive. So this is just the integral portion. And while we add on as a result is R S of X times x plus this plus c so this right here would be our final answer for the integral of SFX Times one the X

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Integration Techniques