jennychan12
  • jennychan12
Find an antiderivative F of f(x) = x^2*sin(x^2) such that F(1) = 0.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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jennychan12
  • jennychan12
I don't think you can use u-sub here. cuz then it'd be f(u) = u*sinu..
jennychan12
  • jennychan12
and i haven't learned trig sub or integration by parts yet.
anonymous
  • anonymous
jenny you are a liar i think

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jennychan12
  • jennychan12
-_- i haven't. but i can do integration by parts. but my teacher never taught it.
anonymous
  • anonymous
becos that problem required fresnal integral
jennychan12
  • jennychan12
what?
anonymous
  • anonymous
Use \(u = x^2\). Then \(x = \sqrt{u}\)
anonymous
  • anonymous
u can use u sub on that
jennychan12
  • jennychan12
\[f(x) = x^2\sin(x^2) \]
jennychan12
  • jennychan12
if u = x^2 then du = 2xdx
anonymous
  • anonymous
well what method are you allowed to use if you can't use trig or parts
jennychan12
  • jennychan12
that's what i don't understand. u-sub doesn't really work...
anonymous
  • anonymous
so ur on u-sub section of the book?
jennychan12
  • jennychan12
no it's a review packet. i learned area, u-sub, trapezoid rule, and simpson's rule
anonymous
  • anonymous
thats the way it is
anonymous
  • anonymous
not an exponent
jennychan12
  • jennychan12
no it's \[f(x) = x^2\sin(x^2)\]
anonymous
  • anonymous
This isn't a candidate for u sub.
anonymous
  • anonymous
u = x^2 du = 2x dx cant use u du sub
anonymous
  • anonymous
no place to apply a substitution
anonymous
  • anonymous
\[ \frac{1}{2}\int \sqrt{u}\sin(u)du \]Isn't helpful.
anonymous
  • anonymous
|dw:1357267172296:dw|
jennychan12
  • jennychan12
-_- it was in the textbook in the chapter 5. we don't do trig sub/parts until chapter 8...
anonymous
  • anonymous
whats the name of the textbook
jennychan12
  • jennychan12
although i know how to do parts but my teacher never taught it.
jennychan12
  • jennychan12
stewart
anonymous
  • anonymous
7th edition?
jennychan12
  • jennychan12
5th
anonymous
  • anonymous
which chapter section?
jennychan12
  • jennychan12
chapter 5 review #49
anonymous
  • anonymous
integral x^2 sin(x^2) dx = 1/4 (sqrt(2 pi) C(sqrt(2/pi) x)-2 x cos(x^2))+constant C(x) is the fresnal C integral Medal pls
anonymous
  • anonymous
fresnal function is introduced in section 5.3
anonymous
  • anonymous
lol wolframalpha? xD
jennychan12
  • jennychan12
haha same 5.3 is fundemental theorum
anonymous
  • anonymous
go to 5.3 examlpe number 3
jennychan12
  • jennychan12
we don't use stewart. we use larson. my teacher just photocopied this from the stewart textbook
anonymous
  • anonymous
thats illegail
anonymous
  • anonymous
but u gonna have to look up fresnal function elsewhere then
jennychan12
  • jennychan12
she owns like ten stewart textbooks. -_- i'll just use parts then...
anonymous
  • anonymous
I guess one way to approach it is: \[F(t) = \int\limits_{1}^{x} t ^{2} \sin(t ^{2})\] This way the integral evaluated at x = 1 results in 0, and its derivative is just the original function
anonymous
  • anonymous
wait let me rewrite that
anonymous
  • anonymous
\[F(x) = \int\limits\limits_{1}^{x} t ^{2} \sin(t ^{2}) dt\]
anonymous
  • anonymous
That way when it is evaluated at F(1), you still get 0. And the derivative is the original function.
anonymous
  • anonymous
If you used Simpson's rule or Midpoint rule on that, you'd get a nasty summation but it is within the rules.
jennychan12
  • jennychan12
oh wow. thanks @LogicalApple that's the answer in the book -___-
anonymous
  • anonymous
Well it makes sense, right? We don't actually have to take an integration. We know from the fundamental theorem of calculus (either part 1 or 2, i forget) that the derivative of F(x) here is just the inside of the integral evaluated at x. And for the limits of integration, we know that if we integrate from a to a we always get 0. So let a = 1, that way F(1) is an integral evaluated from 1 to 1.
jennychan12
  • jennychan12
yes but i never thought of it that way... *sigh*
anonymous
  • anonymous
Stewart doesn't mess around.. I'm in chapter 5 on the 7th edition.
jennychan12
  • jennychan12
oh same here i think. my teacher uses both stewart and larson. we teach from larson.
anonymous
  • anonymous
I will have to check that one out. Anyway good luck with your studies!
jennychan12
  • jennychan12
thanks :)
anonymous
  • anonymous
I think it's sort of an unfair question. Should have specified a bit more the rigor.
jennychan12
  • jennychan12
same. my teacher likes stewart because they have "more pictures" and "good problems". which means that she puts them on quizzes and tests. :[
jennychan12
  • jennychan12
i guess it just makes u think about the concept more?
anonymous
  • anonymous
If they said something like "in terms of an indefinite integral" then it would have been fair.
anonymous
  • anonymous
Yeah sometimes Stewart's questions are very very chapter-specific That's why I asked what chapter the problem was in to get an idea of the context of the question.

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