Integral of a sqrt of polynomial

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Homework Help Overview

The discussion revolves around the integral of the function (1/sqrt(x^2+16)) from 0 to 4, involving a trigonometric substitution where x is expressed as 4tan(theta). Participants are exploring the integration process and the implications of changing variables.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the substitution method and its impact on the integral, questioning the simplification of the integrand and the necessity of changing integration limits. There is also confusion regarding whether to revert back to the original variable after integration.

Discussion Status

Some participants have provided guidance on simplifying the integrand and changing limits appropriately. There is an ongoing exploration of the correctness of the results obtained, with no explicit consensus reached yet.

Contextual Notes

Participants are navigating the specifics of definite integrals and the implications of variable substitution, with some expressing uncertainty about the correctness of their results and the necessity of reverting to the original variable.

ldbaseball16
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Homework Statement


(1/sqrt(x^2+16), x, 0, 4);


Homework Equations


x=4tan(theta)
dx=4sec^2(theta)d(theta)


The Attempt at a Solution


(1/sqrt(16+(4tan(theta))^2)(4sec^2theta) I am confused can i get some help??
 
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ldbaseball16 said:

Homework Statement


(1/sqrt(x^2+16), x, 0, 4);


Homework Equations


x=4tan(theta)
dx=4sec^2(theta)d(theta)


The Attempt at a Solution


(1/sqrt(16+(4tan(theta))^2)(4sec^2theta) I am confused can i get some help??

If you remember that

[tex]1 + tan^2 \theta = sec^2 \theta[/tex]

then you'll find that the integrand simplifies quite nicely. Don't forget to change your integration limits appropriately.
 


ok, i got ln(4/sqrt(x^2 +16) + (x/4)? is this right?
 


hmmm... if this is a definite you don't need to try and convert back to x's, just change the integration limits when you make the variable change
 


ldbaseball16 said:
ok, i got ln(4/sqrt(x^2 +16) + (x/4)? is this right?

I think that's pretty close. As the previous post mentions, I didn't change back to the x's but worked with theta limits of 0 to pi/4.
 

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