Integral of tangent squared of x

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SUMMARY

The integral of tangent squared of x, represented as ∫tan²(x) dx, can be solved using the identity 1 + tan²(x) = sec²(x). This relationship simplifies the integration process since the derivative of tan(x) is sec²(x). By recognizing this connection, one can effectively integrate tan²(x) by substituting it with sec²(x) in the integral. The discussion highlights the importance of understanding trigonometric identities for solving integrals involving tangent functions.

PREREQUISITES
  • Understanding of trigonometric identities, specifically 1 + tan²(x) = sec²(x)
  • Knowledge of basic calculus, particularly integration techniques
  • Familiarity with the derivatives of trigonometric functions, especially tan(x) and sec(x)
  • Experience with integral notation and manipulation
NEXT STEPS
  • Study the derivation and applications of the identity 1 + tan²(x) = sec²(x)
  • Practice integrating various trigonometric functions, focusing on sec²(x) and tan²(x)
  • Explore advanced integration techniques, such as integration by substitution
  • Learn about the applications of integrals in physics and engineering contexts
USEFUL FOR

Students and educators in calculus, mathematicians focusing on trigonometric integrals, and anyone interested in enhancing their integration skills involving tangent and secant functions.

grief
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I tried and tried and I'm not able to solve this. I've managed to find the integral of sin squared of x by using the fact that cos(2x)=1-2(sin(x))^2, but I'm not able to do the same for tangent because I'm stuck with a quotient.
 
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Have you ever seen the term tan^2(x) in a derivative of some function? Recognizing this may put you on the right track.
 
well since
1 + (tan(x))^2 = (sec(x))^2

and we know that the derivative of tan(x) is (sec(x))^2

then it's easy to find the integral of (tan(x))^2
 
I've merged your two threads.

P.S.: What "quotient" did you get that you were having trouble integrating? If I could see it, maybe I could give a hint on how to integrate it.

P.P.S.: people learn better when you give them hints, or direction on the problem than when you do most of the steps for them and just leave a short blank at the end.
 
Last edited:
never mind, I was on the wrong track. What d_leet said was right, you need to use (sec(x))^2
 
never mind, I was on the wrong track.
I'm not so sure. You certainly weren't on the easy track, but I am not yet ready to believe you were on the wrong track.
 

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