crazco said:The Attempt at a Solution
let
u = arc cos
du = -1 / (sqrt 1-x^2)
dv = 2x
v= x^2
arc cos * x^2 - the integral of -x^2 / (sqrt 1-x^2)
then i don't know what to do
Integration using separation of parts is a method of solving integrals that involves breaking down a complex function into simpler parts and integrating each part separately.
Integration using separation of parts is used when the function being integrated is a product of two simpler functions, and when the integral cannot be solved using other methods such as substitution or integration by parts.
The method works by using the product rule of differentiation in reverse. The function is broken down into two parts, and each part is then integrated separately. The final solution is the sum of the integrals of the two parts.
The steps for integration using separation of parts are as follows:
1. Identify the function to be integrated as a product of two simpler functions.
2. Choose one of the two functions to be the "u" term and the other to be the "dv" term.
3. Differentiate the "u" term and integrate the "dv" term.
4. Substitute the values in the formula for integration by parts.
5. Evaluate the integral and simplify the result if possible.
Some common mistakes to avoid in integration using separation of parts are:
- Choosing the wrong "u" and "dv" terms
- Forgetting to differentiate the "u" term
- Forgetting to integrate the "dv" term
- Making errors in substitution or simplification
It is important to carefully follow the steps and double-check the solution to avoid these mistakes.