- #1
kasse
- 384
- 1
What is the best way?
What are you talking about?kasse said:not cosine, sine
Try again later :)kasse said:Don't know, I guess I'm too drunk to do maths right now.
So how about just u= cos(x)?kasse said:Oh, I wrote cos^3 x instead of sin^3 x in the headline. That explains my confusion.
sin^3 x
=
sin^2 x*sin x
=
(1 - cos^2 x)sin x
Then substitution?
u = 1-cos^2 x
du/dx = 2cos x*sin*x
so that
sin^3 x dx = - u / 2cos x
Hm...
The integral of cos³(x) refers to the process of finding the antiderivative or the area under the curve of the function cos³(x). This is a trigonometric integral, and solving it involves using techniques specific to integrating powers of trigonometric functions.
To integrate cos³(x), one common technique is to use the trigonometric identity cos²(x) = 1 - sin²(x) and express cos³(x) as cos(x) * (1 - sin²(x)). This transformation allows the integral to be approached with substitution or by breaking it into simpler parts that can be integrated separately.
Yes, u-substitution is a viable method after applying the trigonometric identity. By setting u = sin(x), the integral becomes easier to handle, converting the problem into an integral in terms of u, which can be integrated using standard techniques.
An alternative method involves using the power-reduction formula, which reduces the power of trigonometric functions. For cos³(x), the formula cos²(x) = (1 + cos(2x))/2 can be used. This approach converts the integral into a form that combines linear and quadratic trigonometric terms, which are easier to integrate.
Integration by parts can be used, although it's more complex for cos³(x) compared to other methods. This technique involves integrating cos(x) and differentiating cos²(x), or vice versa. It may require more than one application of integration by parts and often leads to a longer process.
When integrating cos³(x), it's important to be mindful of the identities used and the transformations applied. Each step should simplify the integral. Also, when using substitution, remember to convert all the variables back to the original variable (x) after integrating.