Simplifying the Integral of 20 sec^3(x) dx using Trigonometric Identities

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SUMMARY

The discussion focuses on simplifying the integral of 20 sec3(x) dx using trigonometric identities. The initial transformation involves rewriting the integral as 20 sec2(x) * sec(x) dx, followed by substituting sec2(x) with tan2(x) - 1. The participants explore various substitution methods, including u = tan(x) and u = sec(x), but encounter difficulties in simplification. A successful approach is suggested by multiplying by cos(x)/cos(x) to facilitate integration.

PREREQUISITES
  • Understanding of trigonometric identities, specifically secant and tangent functions.
  • Familiarity with integration techniques, particularly integration by substitution.
  • Knowledge of manipulating integrals involving trigonometric functions.
  • Basic proficiency in calculus, particularly integral calculus.
NEXT STEPS
  • Study the method of integration by substitution in trigonometric integrals.
  • Learn about the application of trigonometric identities in integral calculus.
  • Explore advanced integration techniques, including integration by parts.
  • Practice solving integrals involving secant and tangent functions, focusing on different substitution methods.
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Students and educators in calculus, mathematicians focusing on integral calculus, and anyone seeking to enhance their skills in solving trigonometric integrals.

dexstarr
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I'm having a hard time breaking this down using tri identities.

} = integral sign 20}sec^3(x) dx
20}sec^2(x) * sec(x) dx
20}[tan^2(x)-1] * sec(x) dx

after this I'm stuck. I tried letting u = tan(x) or sec(x) but i can't seem to cancel anything out.
 
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Re: Integral of 20 sec^3(x)dx

dexstarr said:
I'm having a hard time breaking this down using tri identities.

} = integral sign 20}sec^3(x) dx
20}sec^2(x) * sec(x) dx
20}[tan^2(x)-1] * sec(x) dx

after this I'm stuck. I tried letting u = tan(x) or sec(x) but i can't seem to cancel anything out.

$$20 \int \frac{1}{\cos^3(x)}$$

Multiply by $$\dfrac{\cos(x)}{\cos(x)}$$

$$20 \int \frac{\cos(x)}{\cos^4(x)}$$

$$20 \int \frac{\cos(x)}{(1-\sin^2(x))^2}$$

Now use $$u = \sin(x)$$
 
Re: Integral of 20 sec^3(x)dx

SuperSonic4 said:
Multiply by $$\dfrac{\cos(x)}{\cos(x)}$$
Coooool! I've never seen that method.

-Dan
 

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