the integral of 1x2-5x+14/(x-1)(x^2+9)
the answer i got was ln(x-1)-5/3 tan^-1(x/3) however this is not right pleasee help!
To integrate the given function 1x^2 - 5x + 14 / (x - 1)(x^2 + 9), you can use partial fraction decomposition and integration techniques. Here's a step-by-step explanation of how to solve it:
1. Factorize the denominator. The denominator has two distinct linear factors: (x - 1) and a quadratic factor (x^2 + 9).
2. Rewrite the given expression using partial fraction decomposition. Let's assume the expression can be written as A/(x - 1) + (Bx + C)/(x^2 + 9), where A, B, and C are constants.
3. Clear the fractions by finding a common denominator. Multiply both sides of the equation by (x - 1)(x^2 + 9):
1x^2 - 5x + 14 = A(x^2 + 9) + (Bx + C)(x - 1)
4. Expand the equation and combine like terms:
1x^2 - 5x + 14 = Ax^2 + 9A + Bx^2 - Bx + Cx - C
Group the similar terms:
1x^2 - 5x + 14 = (A + B)x^2 + (-B + C)x + (9A - C)
By comparing the coefficients of the powers of x on both sides of the equation, you can form a system of linear equations:
A + B = 1 (for x^2 term)
-B + C = -5 (for x term)
9A - C = 14 (for constant term)
5. Solve the system of linear equations to find the values of A, B, and C. In this case, the solution is: A = 3/10, B = 7/10, and C = -2.
6. Rewrite the original integral using the partial fractions:
∫[1x^2 - 5x + 14 / (x - 1)(x^2 + 9)] dx = ∫[3/10(x - 1) + (7/10x - 2)/(x^2 + 9)] dx
7. Integrate each term separately:
∫[3/10(x - 1) + (7/10x - 2)/(x^2 + 9)] dx = 3/10 ∫(x - 1) dx + 7/10 ∫(x/(x^2 + 9)) dx - 2/10 ∫(1/(x^2 + 9)) dx
Simplify each integral:
= 3/20(x^2 - x) + 7/10 ln|x^2 + 9| - 2/3 arctan(x/3) + C
Therefore, the correct integral of 1x^2 - 5x + 14 / (x - 1)(x^2 + 9) is 3/20(x^2 - x) + 7/10 ln|x^2 + 9| - 2/3 arctan(x/3) + C, where C is the constant of integration.