Please help me integrate this equation using partial fractions:
Integrate [(x^2+5)/(x^3-x^2+x+3)]dx.
Thank you very much.
x^3-x^2+x+3=(x+1)(x^2-2x+3)
(x^2+5)/(x^3-x^2+x+3)=
=A/(x+1)+(Bx+C)/(x^2-2x+3)
A(x^2-2x+3)+(Bx+C)(x+1)=x^2+5
A+B=1
-2A+B+C=0
3A+C=5
Find A,B,C
In google type:
wolfram alpha
When you see list of results click on:
Wolfram Alpha:Computational Knoweledge Engine
When page be open in rectangle type:
(x^2+5)/(x^3-x^2+x+3) and click option =
After few secons you will see all about that function.
Now click option:
Partial fraction expansion: Show steps
Now in wolfram alpha rectangle type:
2/(x^2-2x+3)+1/(x+1)
and click option =
When you see results clic option:
Indefinite integral: Show steps
Thank you very much.
To integrate the given equation using partial fractions, you need to first factor the denominator and then express the integrand as a sum of simpler fractions. Here are the steps to follow:
Step 1: Factor the denominator
To factor the denominator x^3 - x^2 + x + 3, you can use various methods such as synthetic division or the rational root theorem. In this case, the factorization is (x - 1)(x + 1)(x + 3).
Step 2: Express the integrand as a sum of partial fractions
Now, express the integrand [(x^2 + 5)/(x^3 - x^2 + x + 3)] as a sum of partial fractions. Let A, B, and C be constants to be determined.
[(x^2 + 5)/(x^3 - x^2 + x + 3)] = A/(x - 1) + B/(x + 1) + C/(x + 3)
Step 3: Find the values of A, B, and C
To find the values of A, B, and C, you need to equate the numerators of both sides of the equation.
x^2 + 5 = A(x + 1)(x + 3) + B(x - 1)(x + 3) + C(x - 1)(x + 1)
Expand the right side and collect similar terms to obtain:
x^2 + 5 = (A + B + C)x^2 + (4A + 2B - 2C)x + (3A - 3B + A - B)
Now, equate the coefficients of the corresponding powers of x:
1. Coefficient of x^2: A + B + C = 1
2. Coefficient of x: 4A + 2B - 2C = 0
3. Coefficient of constant (x^0): 3A - 3B + A - B = 5
Solving this system of linear equations will give you the values of A, B, and C.
Step 4: Find the values of A, B, and C (continued)
To solve the system of linear equations, you can use any method you prefer, such as substitution or elimination.
Using the elimination method, we can solve the equations as follows:
1. From equation (2), we can express A in terms of B and C:
4A + 2B - 2C = 0 ---> A = (2C - B)/2
2. Substituting A into equations (1) and (3), we get:
(2C - B)/2 + B + C = 1 ---> C = 1/4
3(2C - B)/2 - 3B + (2C - B) - B = 5 ---> B = 3
Now, substitute the values of B and C into A:
A = (2(1/4) - 3)/2 = -5/4
Step 5: Rewrite the integrand using the partial fraction decomposition
Rewrite the integrand [(x^2 + 5)/(x^3 - x^2 + x + 3)] as a sum of partial fractions:
[(x^2 + 5)/(x^3 - x^2 + x + 3)] = -5/(4(x - 1)) + 3/(x + 1) + 1/(4(x + 3))
Step 6: Integrate the partial fractions
Now, integrate each of the partial fractions separately. The integral of a constant is equal to the constant multiplied by x:
∫[-5/(4(x - 1)) + 3/(x + 1) + 1/(4(x + 3))] dx
= (-5/4)∫(1/(x - 1)) dx + 3∫(1/(x + 1)) dx + (1/4)∫(1/(x + 3)) dx
The integrals of the individual fractions are straightforward to evaluate using logarithmic and inverse trigonometric functions.
Step 7: Evaluate the integrals and simplify the result
∫(1/(x - 1)) dx = ln|x - 1|
∫(1/(x + 1)) dx = ln|x + 1|
∫(1/(x + 3)) dx = ln|x + 3|
Now, substitute the limits of integration (if given) and subtract the lower limit from the upper limit to get the final result.
Note: If you're not given any limits, you can write the final answer as a sum of the integrals with arbitrary constants, indicating that it's the most general solution.
I hope this explanation helps you integrate the given equation using partial fractions. Let me know if you have any further questions!