How would i go about using partial fraction decomposition on the integral, (int) (x+2)/(x^2+4x+5) dx.
Is there a way to do this utilizing complex number decomposition.
Unfortunately I have to make sure that the normal computation and the complex computation match up
To use partial fraction decomposition, follow these steps:
Step 1: Factorize the denominator:
The denominator of the given integral, x^2 + 4x + 5, cannot be factored into linear terms with real coefficients. Therefore, it does not factor nicely.
Step 2: Complete the square:
To proceed, complete the square in the denominator:
x^2 + 4x + 5 = (x^2 + 4x + 4) + 1 = (x + 2)^2 + 1
Step 3: Write the integral in partial fraction form:
Now, we rewrite the original integral as a sum of simpler fractions using partial fraction decomposition:
(int) (x+2)/(x^2+4x+5) dx = A/(x + 2) + B((x + 2)^2 + 1)
Step 4: Determine the unknown constants:
To find constants A and B, we need to find a common denominator for the partial fractions. Since the denominators are (x + 2) and (x + 2)^2 + 1, the common denominator is (x + 2)^2 + 1.
(int) (x+2)/(x^2+4x+5) dx = A((x + 2)^2 + 1)/((x + 2)^2 + 1) + B(x + 2)/((x + 2)^2 + 1)
Step 5: Combine the fractions and equate coefficients:
Now, we combine the fractions:
(int) (x+2)/(x^2+4x+5) dx = (A((x + 2)^2 + 1) + B(x + 2))/((x + 2)^2 + 1)
Since this equation holds for all values of x, we can equate coefficients of like terms on both sides.
The coefficient of (x + 2)^2 in the numerator on the right side is A, and the coefficient of (x + 2) is A + B.
(x + 2): A + B = 1
(x + 2)^2: A = 0
From the equation A = 0, we find A = 0.
From the equation A + B = 1, we find B = 1.
Step 6: Substitute constants back into the integral:
Now that we have determined A and B, we can substitute them back into the original integral:
(int) (x+2)/(x^2+4x+5) dx = 0 + 1/((x + 2)^2 + 1)
So the integral can be written as:
(int) (x+2)/(x^2+4x+5) dx = 1/((x + 2)^2 + 1)
To answer your second question, complex number decomposition is useful when factoring a quadratic expression with complex roots. However, in this case, the quadratic expression x^2 + 4x + 5 does not have real or complex roots that can be easily factored. Therefore, we cannot utilize complex number decomposition in this particular situation.
You don't want to dive into this with complex numbers. Partial fractions will not help here.
Let
u = x^2+4x+5
du = 2(x+2) dx
and that is
∫ 1/2 du/u = 1/2 lnu = 1/2 ln(x^2+4x+5) + C