Solving Integral of cos^2*sqrt(u)

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SUMMARY

The discussion focuses on solving the integral of cos²(θ) * √(1 + tan²(θ)). Participants clarify that the substitution u = 1 + tan²(θ) leads to complications and is not the optimal approach. Instead, they recommend utilizing trigonometric identities to simplify the integrand before proceeding with integration. The consensus is that recognizing the identity 1 + tan²(θ) = sec²(θ) is crucial for simplifying the problem effectively.

PREREQUISITES
  • Understanding of basic calculus concepts, particularly integration.
  • Familiarity with trigonometric identities, specifically 1 + tan²(θ) = sec²(θ).
  • Knowledge of substitution methods in integration.
  • Ability to differentiate trigonometric functions.
NEXT STEPS
  • Study trigonometric identities and their applications in integration.
  • Learn about integration techniques involving substitution and simplification.
  • Explore examples of integrals involving secant and tangent functions.
  • Practice solving integrals that require recognizing and applying trigonometric identities.
USEFUL FOR

Students studying calculus, particularly those focusing on integration techniques and trigonometric functions.

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Homework Statement


integral of cos^2 * (sqr(1+tan^2))


Homework Equations





The Attempt at a Solution


let u = 1 +tan^2
du = sec^2

im not sure how 1/cos^2 gets accounted for the cos^2 in the front
 
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You did your differentiation wrong. In fact you don't need to use substitution. Firstly what is 1+tan^{2}x?
 
Punkyc7 said:

Homework Statement


integral of cos^2 * (sqr(1+tan^2))


Homework Equations





The Attempt at a Solution


let u = 1 +tan^2
du = sec^2
If u = 1 + tan^2(theta), du = 2tan(theta)*sec^2(theta)d(theta)

In any case, this is not a good substitution to use.

There are some trig identities that can be used to simplify the integrand first, and then the integration is very simple.
Punkyc7 said:
im not sure how 1/cos^2 gets accounted for the cos^2 in the front
 

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