the integral of 1x^2-5x+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!
If your expression was
(x^2-5x+14)/(x-1)/(x^2+9)
The answer ln(x-1) - (5/3)tan-1(x/3)is correct.
Read the instructions carefully, sometimes they require a simplified answer, sometime no factoring is permitted, etc.
To avoid confusion, always put sufficient parentheses and / operators.
The posted expression interpreted algebraically is equivalent to:
x^2 - 5x + (14)/(x-1) * (x^2+9)
which is quite different from the expression above.
im sorry the question was
x^2 - 5x + (14)/(x-1) * (x^2+9)
but my answer was not correct for it, what mistake did i make?
For
x^2 - 5x + (14)/(x-1) * (x^2+9)
there is only (x-1) in the denominator, and consequently, you will have a ln(x-1) term together with a number of polynomial terms.
Check carefully the format of the question. If the question shows a division sign, there are implicit parentheses around the numerator and the denominator that you have to insert before posting the expression.
this is how the question is written:
(x^2 - 5x + 14 )/(x-1) * (x^2+9)
for the answer ln(x-1)-5/3 tan^-1(x/3)is it wrong or do i just need to add parenthesis somewhere?
ln(x-1) - (5/3)tan-1(x/3) is correct for (x^2-5x+14)/(x-1)/(x^2+9) only.
You will have to do the integration again for (x^2 - 5x + 14 )/(x-1) * (x^2+9).
However, it will most probably be easier than before.
To determine the integral of the function, we need to use partial fraction decomposition. Here's how you can proceed:
1. Factor the denominator, which is (x-1)(x^2+9).
It cannot be factored further since x^2+9 is irreducible over the real numbers.
2. Write the fraction as a sum of partial fractions with undetermined coefficients:
A/(x-1) + (Bx + C)/(x^2+9)
3. To find the values of A, B, and C, multiply both sides by the denominator:
1x^2 - 5x + 14 = A(x^2 + 9) + (Bx + C)(x - 1)
4. Expand and equate the coefficients of the like terms:
1x^2 - 5x + 14 = (A + B)x^2 + (C - A)x + (9A - B)
Equating the coefficients, we get the following system of equations:
A + B = 1 (coefficient of x^2)
C - A = -5 (coefficient of x)
9A - B = 14 (constant term)
5. Solve the system of equations to find the values of A, B, and C.
By solving the system, you should find: A = -2/3, B = 1/3, and C = -13/3.
6. Now that you have the partial fractions, you can integrate each term separately.
∫ (1x^2 - 5x + 14)/((x-1)(x^2+9)) dx =
∫ (-2/3)/(x-1) dx + ∫ (1/3)(x-1)/(x^2 + 9) dx + ∫ (-13/3)/(x^2 + 9) dx
7. The integral of -2/3 divided by (x-1) can be easily computed. It is -2/3 ln|x-1| + C1, where C1 is the constant of integration.
8. To integrate (1/3)(x-1)/(x^2 + 9), use u-substitution.
Let u = x^2 + 9, then du = 2x dx, and the integral becomes:
(1/6) ∫ (1/u) du = (1/6) ln|u| + C2
Substitute back u = x^2 + 9:
(1/6) ln|x^2 + 9| + C2
9. Finally, the integral of -13/3 divided by (x^2 + 9) can be computed using arctangent substitution.
Let u = 3x, then du = 3 dx. The integral becomes:
(-13/9) ∫ (1/(u^2 + 9)) du = (-13/9) (1/3) arctan(u/3) + C3
Substitute back u = 3x:
(-13/9) (1/3) arctan(x/3) + C3
10. To obtain the final answer, sum up all the individual integrals:
-2/3 ln|x-1| + (1/6) ln|x^2 + 9| - (13/27) arctan(x/3) + C
So the correct answer for ∫ (1x^2 - 5x + 14)/((x-1)(x^2+9)) dx is:
-2/3 ln|x-1| + (1/6) ln|x^2 + 9| - (13/27) arctan(x/3) + C, where C is the constant of integration.