How Do You Integrate sin^2(t)cos^4(t) from 0 to Pi?

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SUMMARY

The integral of the function \(\int_{0}^{\pi} \sin^{2}(t) \cos^{4}(t) \, dt\) can be solved using trigonometric identities, specifically double angle identities. The discussion emphasizes the need to express \(\sin^{2}(t)\) and \(\cos^{4}(t)\) in terms of \(\cos(2t)\) and \(\sin(2t)\) to simplify the integration process. Participants noted that reversing the limits of the integral is necessary for proper evaluation. The approach involves transforming the integral into a more manageable form using identities like \(2\sin(t)\cos(t) = \sin(2t)\) and \(2\cos^{2}(t) - 1 = \cos(2t)\).

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  • Familiarity with definite integrals and their properties
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  • Research how to apply double angle identities in integration
  • Learn about the reduction formulas for trigonometric integrals
  • Explore techniques for reversing limits in definite integrals
  • Study the relationship between sine and cosine functions through their identities
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Students studying calculus, particularly those focusing on integration techniques, as well as educators seeking to clarify the application of trigonometric identities in solving integrals.

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Homework Statement



\int^{0}_{pi}(sin^{2}t)*(cos^{4}t)

Homework Equations





The Attempt at a Solution



I know you have to use trig identities.. but everything I try does not work.
 
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The limits of the integral should be reversed.
 
skyturnred said:

Homework Statement



\int^{0}_{pi}(sin^{2}t)*(cos^{4}t)

Homework Equations





The Attempt at a Solution



I know you have to use trig identities.. but everything I try does not work.

You can do it with double angle identities. But you haven't shown anything you've tried yet. What did it occur to you to try?
 
skyturnred said:

Homework Statement



\displaystyle \int_{0}^{\pi}(\sin^{2}t)*(\cos^{4}t) dt

Homework Equations





The Attempt at a Solution



I know you have to use trig identities.. but everything I try does not work.

2\sin(\theta)\cos(\theta)=\sin(2\theta)

So, sin2(θ)cos2(θ) =   ?  

2\cos^2(\theta)-1=\cos(2\theta)

So, cos2(θ) =   ?  
 
cos^2(t)=1-sin^2(t), then you use the double cos identity until you reduce it down to combination of cos(2t)'s,
 
Last edited:

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