Integrating Trig Functions: Solving Trig Integration Homework

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Homework Help Overview

The discussion revolves around integrating trigonometric functions, specifically focusing on the integration of expressions involving tangent and secant functions. The original poster presents a series of integration steps and transformations related to the function I(tan^5x,x).

Discussion Character

  • Mathematical reasoning, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • Participants explore various integration techniques and transformations, including the use of substitution and trigonometric identities. There are attempts to clarify steps and identify potential errors in the integration process.

Discussion Status

Some participants have pointed out possible errors in the original poster's calculations, while others suggest using trigonometric identities to simplify the problem. The discussion is ongoing, with multiple interpretations of the integration steps being explored.

Contextual Notes

There is mention of a discrepancy between the original poster's results and those provided in a textbook, indicating a potential misunderstanding or misapplication of integration techniques. The discussion also highlights the importance of correctly applying trigonometric identities.

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Homework Statement



I(tan^5x,x)=I(tan^2xtan^3x,x)=I((sec^2x-1)tan^3x,x)
=I(sec^2xtan^3x,x)-I(tan^3x,x)
u=tanx du=sec^2x
=I(u^3,u)-I(tan^3x,x)
=u^4/4-I(tan^3x,x)
=tan^4x/4-I(tan^2xtanx,x)
=tan^4x/4-I((sec^2x-1)tanx,x)
=tan^4x/4-I(tanxsec^2x-tanx)
=tan^4x/4-I(tanxsec^2x,x)-I(tanx,x)
y=tanx, dy=sec^2x
=tan^4x/4-I(u,u)-I(tanx,x)
=tan^4x/4-u^2/2-ln|secx|+c
tan^4x/4-tan^2x/2-ln|secx|+c


Homework Equations





The Attempt at a Solution

 
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You have a sign error between lines 8 and 9
nameVoid said:
=tan^4x/4-I(tanxsec^2x-tanx)
=tan^4x/4-I(tanxsec^2x,x)-I(tanx,x)
 
Last edited:
I(tan^5x,x)=I(tan^2xtan^3x,x)=I((sec^2x-1)tan^3x,x)
=I(sec^2xtan^3x,x)-I(tan^3x,x)
u=tanx du=sec^2x
=I(u^3,u)-I(tan^3x,x)
=u^4/4-I(tan^3x,x)
=tan^4x/4-I(tan^2xtanx,x)
=tan^4x/4-I((sec^2x-1)tanx,x)
=tan^4x/4-I(tanxsec^2x-tanx)
=tan^4x/4-I(tanxsec^2x,x)+I(tanx,x)
y=tanx, dy=sec^2x
=tan^4x/4-I(y,y)+I(tanx,x)
=tan^4x/4-y^2/2+ln|secx|+c
tan^4x/4-tan^2x/2+ln|secx|+c

my texts solution is 1/4sec^4x-sec^2x+ln|secx|+c
 
To obtain your text's solution you can simply do what you've been doing all along. That is use the identity \tan^2x=\sec^2x-1.
 
Last edited:

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