Improving Trigonometric Integration

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SUMMARY

The integral \(\int\frac{secx}{(tanx)^2}dx\) can be simplified by rewriting secant and tangent in terms of sine and cosine. Specifically, sec x is expressed as \(1/cos x\) and tan x as \(sin x/cos x\), leading to the transformation of the integral into \(\int \frac{cos x}{sin^2 x} dx\). This method effectively reduces the complexity of the original integral, allowing for easier integration.

PREREQUISITES
  • Understanding of trigonometric identities, specifically secant and tangent functions.
  • Familiarity with integration techniques, particularly substitution methods.
  • Knowledge of how to manipulate trigonometric functions into sine and cosine forms.
  • Basic calculus skills, including integration of rational functions.
NEXT STEPS
  • Study the process of transforming trigonometric integrals into sine and cosine forms.
  • Learn about integration techniques involving trigonometric identities.
  • Explore advanced integration methods, such as integration by parts and partial fractions.
  • Practice solving various trigonometric integrals to enhance problem-solving skills.
USEFUL FOR

Students studying calculus, particularly those focusing on integration techniques, as well as educators looking for effective methods to teach trigonometric integration.

ada0713
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Homework Statement



\int\frac{secx}{(tanx)^2}dx


The Attempt at a Solution


I tried all the u subs u=tanx and u=secx
but neither worked.
Should I used other methods?
Please help me with the start!
 
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ada0713 said:

Homework Statement



\int\frac{secx}{(tanx)^2}dx


The Attempt at a Solution


I tried all the u subs u=tanx and u=secx
but neither worked.
Should I used other methods?
Please help me with the start!

You might try reducing this to sine and cosine. sec x= 1/cos x and tan x = sin x/cos x so this is
\int \frac{1}{cos x}\frac{cos^2 x}{sin^2 x} dx= \int \frac{cos x}{sin^2 x} dx


Anything come to mind now?
 

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