"Explain why neither Substitution nor Integration by Parts..." was answered by students like you

Albarellak

Answered question

2021-08-31

Explain why neither Substitution nor Integration by Parts would work to evaluate integral of $\cos (x^{2}) d x$

Nichole Watt

Skilled2021-09-01Added 100 answers

Possible derivation: $\frac{d}{d x} (\cos (x^{2}))$

Using the chain rule,

$\frac{d}{d x} (\cos (x^{2})) = \frac{d \cos (u)}{d u} \frac{d u}{d x}$ , where $u = x^{2}$ and $\frac{d}{d u} (\cos (u)) = - \sin (u) :$

$= - (\frac{d}{d x} (x^{2})) \sin (x^{2})$

Use the power rule,

$\frac{d}{d x} (x^{n}) = n x^{n - 1}$ , where $n = 2$ . $\frac{d}{d x} (x^{2}) = 2 x :$

Answer:

$= - \sin (x^{2}) 2 x$

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