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?
oops. That should be
(x+2)/(x^2+4x+5) = A/(x - (-2-i)) + B/(x - (-2+i))
well, if you insist on using partial fractions, you want something like
(x+2)/(x^2+4x+5) = A/(x - (2-i)) + B/(x - (2+i))
so, what do you get for that?
To use partial fraction decomposition on the given integral, you need to factor the denominator of the integrand and express the fraction as a sum of simpler fractions. In this case, the denominator is a quadratic expression, so you'll first need to factor it.
The quadratic expression x^2 + 4x + 5 cannot be factored into two linear factors with real coefficients. However, it can be factored using complex numbers. The quadratic formula can be used to find the roots (also known as zeros or solutions) of the quadratic equation.
Step 1: Find the roots of the quadratic equation.
The quadratic equation in this case is x^2 + 4x + 5 = 0.
Using the quadratic formula, x = (-4 ± √(4^2 - 4(1)(5))) / (2(1)).
Simplifying this equation gives us x = (-4 ± √(-4)) / 2.
The square root of a negative number is an imaginary number, so we have complex solutions: x = (-4 ± 2i) / 2.
Simplifying further, we get x = -2 ± i.
Step 2: Express the fraction as a sum of simpler fractions.
Since the quadratic factors as (x + 2 - i)(x + 2 + i), you can write the fraction as:
(x+2)/(x^2+4x+5) = A/(x + 2 - i) + B/(x + 2 + i),
where A and B are constants that we need to determine.
Step 3: Solve for A and B.
To solve for A and B, you can multiply both sides of the equation by the denominator (x^2 + 4x + 5) and simplify. This will allow you to obtain a system of linear equations by equating the numerators on both sides.
(x+2) = A(x + 2 + i) + B(x + 2 - i).
Next, you can expand and collect the terms with matching powers of x on both sides of the equation:
x + 2 = (A + B)x + (A - B)2 - A(i) + B(i).
Now, equate the real parts and imaginary parts:
For the real parts: 1 = (A + B)x + 2(A - B),
For the imaginary parts: 0 = -A(i) + B(i).
Equating the real parts gives you the equation A + B = 1.
Equating the imaginary parts gives you the equation -A + B = 0.
Now, you have a system of two linear equations and two unknowns (A and B). Solving this system of equations will give you the values of A and B.
Step 4: Evaluate the integral.
Once you have determined the values of A and B, you can substitute these into the expression obtained in Step 2:
(x+2)/(x^2+4x+5) = A/(x + 2 - i) + B/(x + 2 + i).
Using partial fraction decomposition, you have transformed the original integral into a sum of integrals of simpler fractions. You can then evaluate these simpler integrals separately.
It is worth noting that the complex number decomposition is used in this case due to the complex roots of the quadratic equation. However, if the quadratic equation had real roots, the partial fraction decomposition could have been done using linear factors.