Mastering Trig Subs: Simplify \int\cos^5(x)dx without the Headache

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In summary, trig substitution is a useful technique in calculus for simplifying integrals involving trigonometric functions. The choice of trigonometric function to substitute depends on the form of the integral and mastering this technique involves understanding basic trig identities, practicing with examples, and having a solid understanding of calculus. To simplify the integral of cos^5(x)dx, one can use the substitution u = cos(x) and solve using basic integration techniques, resulting in an answer involving inverse trigonometric functions.
  • #1
silverdiesel
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These trig subs are killing me.

[tex]\int\cos^5(x)dx[/tex]hints?
 
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  • #2
[tex]\cos^{5} x = \left(1 - \sin^2 x \right)^{2} \cos x[/tex]

Regards,
George
 
  • #3
Thanks George... I am slowly getting the hang of this. I appreciate your help.
 
  • #4
One of the first things you should have learned: if you have sine or cosine to an odd power, factor out one of them to use with the dx.

cos5 x dx= (cos4 x)(cos x dx)
= (cos2 x)2 (cos x dx)= (1- sin2 x)2(cos x dx)

Now, let u= sin x.
 

FAQ: Mastering Trig Subs: Simplify \int\cos^5(x)dx without the Headache

What is trig substitution?

Trig substitution is a technique used in calculus to simplify integrals involving trigonometric functions. It involves substituting a trigonometric function for another variable in the integral, making it easier to integrate.

Why is trig substitution useful?

Trig substitution can be useful for simplifying integrals that would be difficult or impossible to integrate using other methods. It can also help to solve integrals involving trigonometric functions that are not in standard form.

How do you choose which trigonometric function to substitute?

The choice of trigonometric function to substitute depends on the form of the integral. Generally, we choose the trigonometric function that will eliminate the most complicated part of the integral.

What is the process for mastering trig substitution?

The process for mastering trig substitution involves understanding the basic trig identities and how to apply them in integrals, practicing with a variety of examples, and being familiar with common substitution patterns. It also helps to have a solid understanding of basic calculus concepts.

How can I simplify the integral of cos^5(x)dx using trig substitution?

To simplify the integral of cos^5(x)dx, we can use the substitution u = cos(x). This will transform the integral into a simpler form that can be solved using basic integration techniques. The final answer will involve inverse trigonometric functions.

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