The Correct Answer and Explanation is:
The correct answer is:
∫ (x³ + 6x² + 11x + 21) / (x² + x + 4) dx = (1/2)x² + 5x + ln(|x² + x + 4|) + C
Explanation:
This problem requires the integration of a rational function. The first. This division results in a quotient of (x + 5) and a remainder of (2x + 1). Therefore, the original rational function can be rewritten as the sum of the quotient and the remainder divided by the original divisor:
(x³ + 6x² + 11x + 21) / (x² + x + 4) = (x + 5) + (2x + 1) / (x² + x + 4)
Now, we can integrate this new expression term by term:
∫ [(x + 5) + (2x + 1) / (x² + x + 4)] dx = ∫(x + 5) dx + ∫(2x + 1) / (x² + x + 4) dx
The first integral is straightforward. Using the power rule for integration, we get:
∫(x + 5) dx = (1/2)x² + 5x
The second integral, ∫(2x + 1) / (x² + x + 4) dx, can be solved using a u-substitution. Let u = x² + x + 4. Then, the differential du is (2x + 1) dx. This perfectly matches the numerator of our fraction. The integral becomes:
∫(1/u) du = ln|u|
Substituting back for u, we obtain ln|x² + x + 4|.
Finally, we combine all the integrated parts and add the constant of integration, C, to find the complete solution:
(1/2)x² + 5x + ln|x² + x + 4| + C step is to examine the degrees of the polynomials in the numerator and the denominator. The numerator, x³ + 6x² + 11x + 21, has a degree of 3, while the denominator, x² + x + 4, has a degree of 2. Since the degree of the numerator is higher than the degree of the denominator, the fraction is improper. We must use polynomial long division to simplify the integrand before integrating.
Performing the long division of (x³ + 6x² + 11x + 21) by (x² + x + 4) results in a quotient of (x + 5) and a remainder of (2x + 1). Therefore, we can rewrite the original expression as:
(x + 5) + (2x + 1) / (x² + x + 4)
Now, we can integrate this new expression term by term:
∫ [(x + 5) + (2x + 1) / (x² + x + 4)] dx = ∫ (x + 5) dx + ∫ (2x + 1) / (x² + x + 4) dx
The first integral is straightforward:
∫ (x + 5) dx = (1/2)x² + 5x
For the second integral, ∫ (2x + 1) / (x² + x + 4) dx, we can use a u-substitution. Let u be the denominator, so u = x² + x + 4. The derivative of u is du = (2x + 1) dx. This conveniently matches the numerator of the fraction. Substituting u and du into the integral gives:
∫ (1/u) du = ln|u|
Substituting back u = x² + x + 4, we get:
ln|x² + x + 4|
Finally, we combine all the parts and add the constant of integration, C.
(1/2)x² + 5x + ln|x² + x + 4| + C
