Evaluate the integral
INT(x^4+6x^3+5x^2+8)/(x^2+6x+5)dx
Please help.
The integrand is:
f(x) = (x^4+6x^3+5x^2+8)/(x^2+6x+5)
The denominator can be factored as:
x^2+6x+5 = (x+1)(x+5)
This means that f(x) has singularities at x = -1 and x = -5, if the numerator isn't zero there. To find the partial fraction expansion, you can simply calculate the singular behavior near these points, subtract from f(x), leving you with a rational function without any singularities, which is therefore a polynomial (which you can easily find).
This has the advantage of not having to solve equations to find the partial fraction expansion. To find the bahavior near x = -1, you expand the function in powers of (x+1), only the leading term is singular
f(x) = 2/(x+1) + non-singular terms
The expnsion around x = -5 yields:
f(x) = -2/(x+5) + non-singular terms
If we subtract the singular terms from
f(x) we will be left with a rational functuon without any singularities, which is therefore a polynomial.
Let's see what we get:
f(x) - 2/(x+1) + 2/(x+5) =
[x^4+6x^3+5x^2+8 - 2(x+5) + 2 (x+1)]/
[(x+1)(x+5)] =
[x^4 + 6 x^3 + 5 x^2]/[(x+1)(x+5)] =
x^2 (x^2 + 6 x + 5)/[(x+1)(x+5)] =
x^2
So, we have:
f(x) - 2/(x+1) + 2/(x+5) = x^2 ---->
f(x) = x^2 + 2/(x+1) - 2/(x+5)
The integral is thus:
1/3 x^3 + 2 Log|x+1| - 2 Log|x+5| + c
(x^4+6x^3+5x^2+8)/(x^2+6x+5)
= x^2 + 8/(x^2 + 6x+5)
= x^2 + 8/((x+1)(x+5))
let 8/((x+1)(x+5) = A/(x+5) + B/(x+1)
8 = A(x+1) + B(x+5)
let x = -1
8 = 4B --> B = 2
let x = -5
8 = -4A
A = -2
so ∫(x^4+6x^3+5x^2+8)/(x^2+6x+5) dx
= ∫ (x^2 - 2/(x+5) + 2(x+1) dx
= (1/3)x^3 - 2ln(x+5) + 2ln(x+1) + c
Did not realize that Count Iblis had already answered this.
I guess it helps to refresh the page before posting.
To evaluate the integral of a rational function, you can use the method of partial fraction decomposition. Here's how you can go about solving this integral step by step:
Step 1: Factorize the denominator.
(x^2 + 6x + 5) can be factored as (x + 5)(x + 1).
Step 2: Write the rational function as a sum of partial fractions.
(x^4 + 6x^3 + 5x^2 + 8) / (x^2 + 6x + 5) = A/(x + 5) + B/(x + 1)
Step 3: Determine the values of A and B.
To find A and B, you can multiply both sides of the equation by the denominator (x^2 + 6x + 5) and then simplify the equations:
(x^4 + 6x^3 + 5x^2 + 8) = A(x + 1) + B(x + 5)
Now, expand and equate the coefficients of like terms on both sides of the equation:
For x^4 term: 1 = A
For x^3 term: 6 = A + B
For x^2 term: 5 = A + 5B
For constant term: 8 = B
From the equations:
A = 1
B = 2
Step 4: Rewrite the integral using the partial fraction decomposition.
The integral becomes:
∫ (x^4 + 6x^3 + 5x^2 + 8)/(x^2 + 6x + 5) dx = ∫ (1/(x + 5) + 2/(x + 1)) dx
Step 5: Evaluate the integral.
Using the result of partial fraction decomposition, integrate each term individually:
∫ (1/(x + 5) + 2/(x + 1)) dx = ∫ (1/(x + 5)) dx + ∫ (2/(x + 1)) dx
The integral of 1/(x + 5) can be evaluated as ln|x + 5| + C1, where C1 is the constant of integration.
The integral of 2/(x + 1) can be evaluated as 2ln|x + 1| + C2, where C2 is the constant of integration.
So the final result is:
∫ (x^4 + 6x^3 + 5x^2 + 8)/(x^2 + 6x + 5) dx = ln|x + 5| + 2ln|x + 1| + C
Note that C is the constant of integration, and it accounts for any additional terms that might have been present in the original expression.