Integration involving trig functions and various powers of X

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Homework Help Overview

The discussion revolves around the integration of a function involving trigonometric and polynomial components, specifically the integral ∫[6x^6 sin(9x)]/[1+x^10] * dx. Participants are exploring substitution methods and the implications of definite integrals.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the use of substitution, with one suggesting setting u = x^6 and questioning the next steps. Others express doubt about the effectiveness of the substitution and emphasize the need to eliminate x from the integral.

Discussion Status

The conversation is ongoing, with participants providing insights into the challenges of the substitution method. One participant has clarified that the integral is definite, which adds complexity to the problem. There is also a discussion about the nature of the integrand, with one participant suggesting it might be an even function.

Contextual Notes

Participants are currently studying definite integrals and substitution techniques. There is a focus on understanding the properties of the integrand, particularly whether it is an even or odd function, which may influence the evaluation of the integral.

seanoe25
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∫[6x^6 sin (9x)]/[1+x^10] * dx

I've set u =x^6
du=6x^5*dx
dx=du/6x^5

∫[6x^6 sin (9x)]/[1+x^10] * (du/6x^5)
=
∫[x*sin(9x)*du]/1+x^10.

Can someone help me figure out the next step? I'm thinking of putting a constant out in front, so I can use 2du for (x^10)
 
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seanoe25 said:
∫[6x^6 sin (9x)]/[1+x^10] * dx

I've set u =x^6
du=6x^5*dx
dx=du/6x^5

∫[6x^6 sin (9x)]/[1+x^10] * (du/6x^5)
=
∫[x*sin(9x)*du]/1+x^10.

Can someone help me figure out the next step? I'm thinking of putting a constant out in front, so I can use 2du for (x^10)
I doubt that this will work. That's a messy integral.

However, you haven't finished the substitution ! You should end up with an integral in the variable, u, with no x what-so-ever.

Are you sure you have written the problem correctly?

What topics are you currently covering in whatever class this problem is from?
 
Yeah, sadly I have written it down properly. Right now we're covering how to solve definite integrals with the use of substitution; it's a beautiful thing when it works, but these problems are moral-breakers. It's acually:

∫ 6x^6 sin(9x)/[1+x^10] *dx

with the upper limit set at pi/2, and the lower limit at -pi/2.

I excluded the limits part because I felt once I got help with the substitution, I had the problem down. But finding the right u is very difficult.
 
seanoe25 said:
Yeah, sadly I have written it down properly. Right now we're covering how to solve definite integrals with the use of substitution; it's a beautiful thing when it works, but these problems are moral-breakers. It's acually:

∫ 6x^6 sin(9x)/[1+x^10] *dx

with the upper limit set at pi/2, and the lower limit at -pi/2.

I excluded the limits part because I felt once I got help with the substitution, I had the problem down. But finding the right u is very difficult.
Having it be a definite integral makes all the difference in the world!

Is the integrand either an even or an odd function?
 
A glimpse of hope! I believe the integrand is an even function. Because when I plugged in f(-x), everything came out to be the same
 
seanoe25 said:
A glimpse of hope! I believe the integrand is an even function. Because when I plugged in f(-x), everything came out to be the same
Not quite!

sin(-9x) = -sin(9x)

Graph the integrand.
 

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