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

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To simplify the integral ∫cos^5(x)dx, it is recommended to factor out one cosine term when dealing with sine or cosine raised to an odd power. This allows for the substitution of u = sin(x), transforming the integral into a more manageable form. The expression can be rewritten as (cos^4(x))(cos(x)dx), which further simplifies to (1 - sin^2(x))^2(cos(x)dx). This method effectively reduces the complexity of the integral. Mastering these trigonometric substitutions can significantly ease the process of integration.
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These trig subs are killing me.

\int\cos^5(x)dxhints?
 
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\cos^{5} x = \left(1 - \sin^2 x \right)^{2} \cos x

Regards,
George
 
Thanks George... I am slowly getting the hang of this. I appreciate your help.
 
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.
 

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