I have to find the integral of 1/(sq. rt. of (4x-x²)) dx. I know I have to complete the square of the denominator, but im confused on how that process works when im only given 4x-x². Could you demonstrate how to complete the square for that equation? Thank you so much!
4x - x^2
-(x^2 - 4x)
x^2 - 4x = 0
x^2 - 4x + 4 = 0 + 4
(x - 2)^2 = 4
(x - 2)^2 - 4
-(x - 2)^2 + 4
4 - (x - 2)^2
| = integral sign
| 1/(sqrt(4 - (x - 2)^2) dx
u = ( x - 2)
du = dx
| 1/(sqrt(4 - u^2)) du
= arcsin u/2 + C
= arcsin (x - 2)/2 + C
Certainly! To complete the square for the expression 4x - x², you'll need to follow these steps:
Step 1: Rearrange the terms
Rearrange the expression so that the x² term comes before the x term: -x² + 4x.
Step 2: Divide by the coefficient of x²
Divide each term of the equation by the coefficient of x² (-1) to make the coefficient 1: x² - 4x.
Step 3: Take half of the coefficient of x and square it
Half of the coefficient of x is -4/2 = -2. Square this value to get (+2)² = 4.
Step 4: Add the result from step 3 to both sides of the equation
Add the result from step 3, which is 4, to both sides of the equation. This keeps the equation equivalent: x² - 4x + 4.
Step 5: Factor the perfect square trinomial
The expression x² - 4x + 4 is now a perfect square trinomial and can be factored as (x - 2)².
Therefore, the completed square for the expression 4x - x² is (x - 2)².
Now, let's proceed to the integration of the given expression using the completed square:
∫(1/√(4x - x²)) dx
Step 1: Substitute the completed square
Substitute the completed square (x - 2)² for the denominator: ∫(1/√((x - 2)²)) dx.
Step 2: Apply the properties of square roots
Since the square root of a perfect square is the positive value of the square root, we can simplify the expression: ∫(1/(x - 2)) dx.
Step 3: Apply the natural logarithm rule
The integral of 1/(x - 2) is the natural logarithm of the absolute value of (x - 2): ln| x - 2 |.
Therefore, the integral of 1/(√(4x - x²)) dx is ln| x - 2 | + C, where C is the constant of integration.
I hope this explanation helps! Let me know if you have any further questions.