Answer:
The evaluated integral is ∫(-4x - 4)/(x^2 + 1) dx = -2ln|x + i| - 2ln|x - i| + C
Explanation:
To evaluate the integral ∫(-4x - 4)/(x^2 + 1) dx, we can use the method of partial fractions.
First, let's rewrite the integrand as a sum of two fractions:
(-4x - 4)/(x^2 + 1) = (-4x)/(x^2 + 1) - 4/(x^2 + 1)
Now, let's find the partial fraction decomposition. We can express the integrand as:
(-4x)/(x^2 + 1) - 4/(x^2 + 1) = A/(x + i) + B/(x - i)
where A and B are constants to be determined. Here, i represents the imaginary unit (√(-1)).
To find A and B, we can multiply both sides of the equation by x^2 + 1 and then equate the numerators:
-4x - 4 = A(x - i) + B(x + i)
Expanding and simplifying:
-4x - 4 = (A + B)x + (B - A)i
Equating the real and imaginary parts separately:
-4x = (A + B)x (equation 1)
-4 = (B - A)i (equation 2)
From equation 1, we have A + B = -4.
From equation 2, we have B - A = 0, which implies B = A.
Substituting B = A into A + B = -4, we get 2A = -4, which gives A = -2. Consequently, B = -2 as well.
Now, we can rewrite the integral using the partial fraction decomposition:
∫(-4x - 4)/(x^2 + 1) dx = ∫(-2/(x + i) - 2/(x - i)) dx
Integrating each term separately:
= -2∫1/(x + i) dx - 2∫1/(x - i) dx
To integrate 1/(x + i), we can use the natural logarithm:
= -2ln|x + i| - 2ln|x - i| + C
where C is the constant of integration.
Therefore,
∫(-4x - 4)/(x^2 + 1) dx = -2ln|x + i| - 2ln|x - i| + C is the integral.