Confusing Integration Example: Solving Problems with Constant Factors of n

[James2]

So, um, I am getting confused on integration problems where you have to do something with "a constant factor of n". Like, this example...

$\int\sqrt{1 + e^{4x^{3}}}e^{4x^{3}}x^{2}dx$

Then the example says to match it to the formula $\int u^{n}du$

Okay... so it does that, but then... something I don't quite understand happens. It says that "du = $e^{4x^{3}}(12x^{2})$" WAIT? WHERE DID THE 12 COME FROM? Then it says that "our integrand contains all of du except for the constant factor of 12" Then it does this...

$\frac{1}{12}\int(1 + e^{4x^{3}})^{1/2} e^{4x^{3}}(12x^{2})dx$

Then it integrates like normal... but... WHERE DID THE 12 COME FROM? I don't know why, it just isn't obvious where this "constant factor of 12" came from?

 

[arildno]

What is the derivative of 4x^3?

 

[James2]

So wait, I have to find the derivative of the exponent then add it to the terms in du? Like, there was an x^2 so I add 12x^2 to that and then take the reciprocal of 12 and move it outside the integral?

 

[James2]

Seriously, somebody, I feel dumb because I am not getting this like am I supposed to find the derivative of u then add/multiply by the other thing?? UHHHHH...

 

[AlephZero]

If you compared the integrals, presumably the example then wants to substitute $u = \sqrt{1 + e^{4x^3}}$

So what does $du/dx$ equal?

 

[arildno]

So wait, I have to find the derivative of the exponent then add it to the terms in du? Like, there was an x^2 so I add 12x^2 to that and then take the reciprocal of 12 and move it outside the integral?

No.
What is the derivative of 1+e^(4x^3)??

 

[James2]

Um I think this is the derivative..

$\frac{du}{dx} = 12x^{2}e^{4x^{3}}$

OH OH OH! I'M SUPPOSED TO TAKE THE DERIVATIVE OF U THEN ADD IT TO THE OTHER PART? Right? And when do I have to move a constant factor outside the integral?

 

[arildno]

Now, I hope you see that 1=1/12*12.

Thus, we recognize that the expression in your original integral, "e^(4x^3)x^2dx"=1/12du"
Agreed?
Furthermore, the square root thing is now to be written as sqrt(u). Agreed?

 

[James2]

Yes, I know 1/12(12)=1 and sure, sqrt(u). Oh wait... du means... derivative of u... okay but what happens to the things that aren't part of u?

 

[arildno]

So, then you have no further problems?

 

[James2]

Well actually one last thing, what happens to stuff that isn't a part of u? Like the x^2 outside of sqrt(u)?

 

[arildno]

Read your own post 7 again.

 

[James2]

Oh wait I don't add it... I replace it? OR DO I ADD IT? Lol I'm confusing myself now.

 

[James2]

Okay I figured it out and worked a practice problem, the answers are at the back of the book but... I got:

$-\frac{(cos2x + 1)^{3/2}}{3/2} + C$

However, the answer to the practices in the back of the book says it is over 3 not 3/2? What happened here, I'm sure it's algebraic but still...

Wait... is it because du = -2sin2x? Then the -2 would be in front and the 2's cancel? Is that right?

 

[Millennial]

How do you expect us to verify your answer without first knowing the initial question?

If your answer is true though, then the question must be this:

$$\int 2\sin(2x)\sqrt{\cos(2x)+1}\,dx$$

which is easily integrable (what is the derivative of $\cos(2x)+1$?)