Explanation:
Factor the denominator to simplify the expression: x^2 + x - 2 = (x - 1)(x + 2)
Use partial fraction decomposition to write the integrand as the sum of simpler terms: (x^3 + 2x^2 + 3x + 4)/(x^2 + x - 2) = A/(x - 1) + B/(x + 2) + Cx + D, where A, B, C, and D are constants.
Multiply both sides of the equation by (x^2 + x - 2) to find the values of A, B, C, and D: x^3 + 2x^2 + 3x + 4 = A(x + 2) + B(x - 1) + (Cx + D)(x^2 + x - 2)
Substitute x = 1 and x = -2 into the equation to find two equations for A, B, C, and D:
x = 1: 4 + 2 + 3 + 4 = 9 = A(-2) + B + (C + D)(-1)
x = -2: -8 - 4 + 6 - 8 = -16 = A(1) + B(-2) + (C - 2D)(4)
Solve the system of equations to find the values of A, B, C, and D:
A = 9/3, B = 5, C = -11/3, D = 7/3
Use the partial fraction decomposition to integrate the integrand:
Integral of 1/(x - 1) = ln|x - 1|
Integral of 1/(x + 2) = ln|x + 2|
Integral of x = x^2/2
Integral of 1 = x
Evaluate the definite integral by subtracting the values of the antiderivatives at the limits of integration:
Integral from x=0 to x=1 of (x^3 + 2x^2 + 3x + 4)/(x^2 + x - 2) = (ln|x + 2| - ln|x - 1| + x^2/2 - x)|x=1 - x=0 = (ln(3) - ln(-1) + 1/2 - 1) = (ln(3) + ln(1) + 1/2) = ln(3) + 1.
So the definite integral of (x^3 + 2x^2 + 3x + 4)/(x^2 + x - 2) from x=0 to x=1 is equal to ln(3) + 1.