for the problem asking to integrate (x^2 - 5x+ 14 )/(x-1) * (x^2+9) I solved getting the answer ln(x-1)-5/3
tan^-1(x/3)however this isn't right.
plsee help!
To integrate the given expression, we can split it into partial fractions:
The expression is (x^2 - 5x + 14) / (x - 1) * (x^2 + 9).
First, let's factorize the denominator as (x - 1) * (x^2 + 9):
(x^2 - 5x + 14) / ((x - 1) * (x^2 + 9)).
To find the partial fractions, we set up the equation:
(x^2 - 5x + 14) / ((x - 1) * (x^2 + 9)) = A / (x - 1) + (Bx + C) / (x^2 + 9).
Next, we find the common denominator:
(x^2 - 5x + 14) = A * (x^2 + 9) + (Bx + C) * (x - 1).
Expanding and simplifying:
x^2 - 5x + 14 = Ax^2 + 9A + Bx^2 - Bx + Cx - C.
Now, we group like terms:
(x^2 - 5x + 14) = (A + B) * x^2 + (-B + C) * x + (9A - C).
By comparing the coefficients of each term on both sides, we get the following system of equations:
A + B = 1 (coefficients of x^2 terms)
-B + C = -5 (coefficients of x terms)
9A - C = 14 (constant terms)
Solving this system of equations will give us the values of A, B, and C.
After solving, we get:
A = 2/3
B = 1/3
C = -17/3
Now we can rewrite the integral as:
∫[(2/3)/(x - 1)] + [(x/3 + 17/3)/(x^2 + 9)] dx.
We can now integrate each term separately using the appropriate rules.
The integral of (2/3) / (x - 1) with respect to x is 2/3 * ln|x - 1| + C1.
The integral of (x/3 + 17/3) / (x^2 + 9) with respect to x is (1/6) * ln| x^2 + 9| + C2.
Therefore, the final result of the integration is:
2/3 * ln|x - 1| + (1/6) * ln|x^2 + 9| + C,
where C is the constant of integration.
Please note that the answer you provided, ln(x-1) - 5/3 tan^-1(x/3), is not correct.