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myininayaBest ResponseYou've already chosen the best response.0
Let F be the antiderivative of \[f(x)=\frac{u^21}{u^2+1}\] Now we have \[[g(x)]'=[F(x)_{2x}^{3x}]=[F(3x)F(2x)]'=(3x)'f(3x)(2x)'f(2x)\]
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
that chain rule stuff is the best! :)
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
so we have \[3 \cdot \frac{(3x)^21}{(3x)^2+1}2 \cdot \frac{(2x)^21}{(2x)^2+1}\]
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
\[3 \cdot \frac{9x^21}{9x^2+1}2 \cdot \frac{4x^21}{4x^2+1}\]
 2 years ago

satellite73Best ResponseYou've already chosen the best response.2
\[\int _a^{2x}\frac{u^21}{u^2+1}du+\int_a^{3x}\frac{u^21}{u^2+1}du\]
 2 years ago

satellite73Best ResponseYou've already chosen the best response.2
dang did that take me a long time to type now chain rule
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
yes it did satellite :)
 2 years ago

satellite73Best ResponseYou've already chosen the best response.2
i am tired, time for bed. gnight
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
me too we are going to bed
 2 years ago

MertsjBest ResponseYou've already chosen the best response.3
\[g'(x)=\frac{2(4x^21)}{4x^2+1}+\frac{3(9x^21)}{9x^2+1}\]
 2 years ago

MertsjBest ResponseYou've already chosen the best response.3
I don't get the 2 and the 3
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
:do you understand what I did?
 2 years ago

MertsjBest ResponseYou've already chosen the best response.3
I thought the u^21/u^2+1 would be the derivative and all I had to do was plug in the interval endpoints.
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
\[g(x)=\int\limits_{2x}^{3x}f(u) du=F(u)_{2x}^{3x}=F(3x)F(2x)\] Do you understand this part?
 2 years ago

MertsjBest ResponseYou've already chosen the best response.3
Why isn't f(u) the g'(x)?
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
no \[g(x)=F(3x)F(2x)\]
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
g is the integral from 2x to 3x of f(u) with respect to u
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
now since we know that \[g(x)=F(3x)F(2x)\] we can differentiate both sides
 2 years ago

MertsjBest ResponseYou've already chosen the best response.3
Ok. I get that and I understand what you wrote about F(3x)F(2x)
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
\[g'(x)=(3x)'f(3x)(2x)'f(2x)\]
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
i used chain rule for this
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
\[g'(x)=3 f(3x)2 f(2x)\] since (3x)'=3 and (2x)'=2
 2 years ago

MertsjBest ResponseYou've already chosen the best response.3
And what happened to F(x)?
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
well (F(3x))'=(3x)'f(3x) by chain rule
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
remember we differentiate both sides so we could find g'
 2 years ago

MertsjBest ResponseYou've already chosen the best response.3
Yep. Ok. I think i can figure it out now. thank you very much and how did you get so smart?
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
i'm a little smart not much
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
well i'm not really smart lol
 2 years ago

MertsjBest ResponseYou've already chosen the best response.3
thanks to you and sleep well...
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
thanks i'm going to sleep for reals i'm so tired by the way satellite likes to do these types of problems another way but i love my way
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
he likes to split the integral at a constant
 2 years ago

myininayaBest ResponseYou've already chosen the best response.0
and then use the other part of the fundamental thm of calculus
 2 years ago

MertsjBest ResponseYou've already chosen the best response.3
Satellite is very smart also. Do you know him personally?
 2 years ago
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