What is the antiderivative of?
(x^2 - 4) / (x - 2)
note: x^2-4=(x-2)(x+2)
simplify the expression
((x-2)(x+2) / (x-2) ] / (x-2)
int x+2 = 1/2 x^2 + 2x + c
To find the antiderivative of a function, we need to reverse the process of differentiation. In this case, we will use a method called partial fraction decomposition.
Step 1: Factorize the denominator
(x^2 - 4) can be factored as (x - 2)(x + 2).
Step 2: Write the fraction as a sum of partial fractions
We can express the given fraction as:
(x^2 - 4) / (x - 2) = (A / (x - 2)) + (B / (x + 2))
Step 3: Find the common denominator
To add or subtract fractions, they need to have a common denominator. In this case, the common denominator is (x - 2)(x + 2).
Step 4: Add the partial fractions
Combining the fractions, we get:
(x^2 - 4) / (x - 2) = (A(x + 2) + B(x - 2)) / (x - 2)(x + 2)
Step 5: Equating coefficients
Comparing the numerators, we have:
x^2 - 4 = (A(x + 2) + B(x - 2))
Step 6: Solve for A and B
Expanding the right side and collecting like terms, we obtain:
x^2 - 4 = (A + B)x + (2A - 2B)
Comparing coefficients, we have two equations:
A + B = 0 (coefficient of x^1)
2A - 2B = -4 (coefficient of x^0)
Solving these equations simultaneously, we find:
A = -2
B = 2
Step 7: Write the partial fractions
We can rewrite the original fraction:
(x^2 - 4) / (x - 2) = (-2 / (x - 2)) + (2 / (x + 2))
Step 8: Evaluate the antiderivative
Now, we can integrate each partial fraction separately:
∫[(x^2 - 4) / (x - 2)] dx = ∫[(-2 / (x - 2)) + (2 / (x + 2))] dx
Integrating each term:
= -2∫(1 / (x - 2)) dx + 2∫(1 / (x + 2)) dx
The antiderivative of 1 / (x - 2) is ln|x - 2|, and the antiderivative of 1 / (x + 2) is ln|x + 2|.
Thus, the antiderivative of (x^2 - 4) / (x - 2) is:
-2ln|x - 2| + 2ln|x + 2| + C, where C is the constant of integration.