zned6559
  • zned6559
help please i need someone to explain Find S d/dx cosx^2 dx=
Mathematics
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katieb
  • katieb
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zned6559
  • zned6559
\[\int\limits_{}^{}\frac{ d }{ dx }cosx^2dx\]
IrishBoy123
  • IrishBoy123
\(\Large I (x) = \int\limits_{}^{}\frac{ d }{ dx }cosx^2 \; dx\) \(\Large = \frac{ d }{ dx } \int\limits_{}^{}cosx^2 \; dx\) the integral and the derivative are "inverses". in the sense that one leads to the other, and vice versa. and this simple idea has a very fancy name: https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus
IrishBoy123
  • IrishBoy123
so to restate: \[\Large \int\limits_{}^{}\frac{ d }{ dx }cosx^2 \; dx = \frac{ d }{ dx } \int\limits_{}^{}cosx^2 \; dx\] or, in plain-speak, the integral of the derivative equals the derivative of the integral.

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zned6559
  • zned6559
so would the answer be cosx^2 since the d/dx and Sdx cancels?
IrishBoy123
  • IrishBoy123
well, sort of or mostly or maybe not...i was being very hand-wavey to get the point across. the bad boy in the room is the integration constant http://www.wolframalpha.com/input/?i=d%2Fdx+%5Bcos%28x%5E2%29%5D http://www.wolframalpha.com/input/?i=integrate+-2x+sin%28x%5E2%29 these are indefinite integrals. it can all be worked around but your question is quite specific so i think that means you need to countenance the integration constant and factor that into your thought process.

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