Definite Integral of a Trig Fxn

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Homework Help Overview

The discussion revolves around evaluating the definite integral of a trigonometric function, specifically ∫tan^5(x) sec(x) dx from 0 to π/3. Participants are exploring methods to manipulate the integrand for easier integration.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to split tan^5(x) into a form involving sec^2(x) but expresses difficulty due to only having sec(x). Another participant suggests transforming the integral into a form involving f(cos(x))sin(x)dx. Additionally, there is a mention of splitting tan^5(x) into tan^4(x)tan(x) and applying a trigonometric identity to facilitate integration.

Discussion Status

The discussion is active, with participants offering different approaches to tackle the integral. One participant indicates that a suggested method was effective, but there is no clear consensus on a single approach yet.

Contextual Notes

Participants are working within the constraints of standard trigonometric identities and integration techniques, and there is an emphasis on finding a suitable form for the integral without providing a complete solution.

johnhuntsman
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π/3
∫tan^5 x sec x
0

I want to split the tan^5 x into a (sec^2 x - 1) but I only have sec x. So I'm a bit stuck. Help is appreciated.
 
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Bring the integral in the form ∫f(cos(x))sin(x)dx.

ehild
 
johnhuntsman said:
π/3
∫tan^5 x sec x
0

I want to split the tan^5 x into a (sec^2 x - 1) but I only have sec x. So I'm a bit stuck. Help is appreciated.
Finding the anti-derivative of tan5(x) sec(x) is a lot like solving your previous tread.

Split tan5(x) into tan4(x) tan(x) then use the trig identity which will change tan4(x)  into a combination of secant functions.
 
^ That did the trick. Thank you : D
 

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