school exam question

[arsene2000]

Let F be the primitive of the function such that, . Calculate the value of
Use radian arcs in your answer.

 

[Deleted member 4993]

Let F be the primitive of the function such that, . Calculate the value of
Use radian arcs in your answer.

Please define "primitive" as defined in your class or textbook.

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem

 

[Dr.Peterson]

Let F be the primitive of the function such that, . Calculate the value of
Use radian arcs in your answer.

It appears that "a primitive of a function" means an antiderivative, that is, a function F whose derivative is f.

We will need to see what you have learned about this process. Do you have a formula that applies to this function (or perhaps a formula for a derivative that looks like it)? Or have you learned to do trigonometric substitutions? What ideas do you have, based on what you have been taught? We don't want to give you help that you can't use.

 

[HallsofIvy]

The standard use of "primitive" in Calculus is what I would call an anti-derivative.
I see that the denominator of this fraction is \(\displaystyle 16+ 81x^2= 4^2+ (9x)^2\) and I recall that the anti-derivative of \(\displaystyle \frac{dx}{x^2+ 1}=arctan(x)+ C\).

So write \(\displaystyle \frac{3}{16+ 81x^2}= \frac{3}{16}\frac{1}{\frac{81}{16}x^2+ 1}= \frac{3}{16}\frac{1}{\left(\frac{9x}{4}\right)^2+ 1}\).

Let \(\displaystyle u= \frac{9x}{4}\) so \(\displaystyle du= \frac{9}{4}dx\), \(\displaystyle dx= \frac{4}{9}du\).

\(\displaystyle f(x)dx= \frac{3}{16}\frac{4}{9}\frac{du}{u^2+ 1}\).

The integral (anti-derivative, primitive) of that is \(\displaystyle \frac{1}{12}arctan(u)= \frac{1}{12} arctan(9x/4)+ C\).

 

[Deleted member 4993]

The standard use of "primitive" in Calculus is what I would call an anti-derivative.

It appears that "a primitive of a function" means an antiderivative, that is, a function F whose derivative is f.

I was hoping that the OP would provide that information to display "primitive" effort towards independent effort!

 

[pka]

Let F be the primitive of the function such that, . Calculate the value of Use radian arcs in your answer.

I was hoping that the OP would provide that information to display "primitive" effort towards independent effort!

Prof. Khan, I seriously doubt that a privative as we mathematicians usually use that term is at all involve here.
Consider \(\displaystyle{F(x)=\int {\frac{3}{{16 + 81{x^2}}}dx}}= \dfrac{1}{12}\arctan\left(\dfrac{9x}{4}\right)\) SEE HERE.
In that case \(F(0)=0\)

 

[Dr.Peterson]

Prof. Khan, I seriously doubt that a privative as we mathematicians usually use that term is at all involve here.
Consider \(\displaystyle{F(x)=\int {\frac{3}{{16 + 81{x^2}}}dx}}= \dfrac{1}{12}\arctan\left(\dfrac{9x}{4}\right)\) SEE HERE.
In that case \(F(0)=0\)

Aren't you forgetting the constant of integration? Surely you know there are infinitely many "primitives" (antiderivatives) for the given function; they asked for the one with a given value of F(0).

 

[pka]

Aren't you forgetting the constant of integration? Surely you know there are infinitely many "primitives" (antiderivatives) for the given function; they asked for the one with a given value of F(0).

I wrote my PhD thesis in integration theory with one of the leading theorist from the texas school. He would not let us use the C, saying that there is no constant of integration. Sorry I just fall easily fall back on old habits.