Find the integral of 2x/(x^2 + 6x + 13)dx
∫ 2xdx/(x^2 + 6x + 13)
Complete the squares of the denominator to get
(x+3)²+2²
Let y = x+3
dy = dx
and x = y-3
I=∫ 2xdx/(x^2 + 6x + 13)
= ∫ 2(y-3)dy/(y²+2²)
=∫ 2ydy/(y²+2²) - ∫6dy/(y²+2²)
The first one integrates to a log, and the second one to an atan, giving
I=ln(x²+6x+13) - 3atan((x+3)/2) + C
To find the integral of 2x/(x^2 + 6x + 13), we can use the method of partial fractions. Follow these steps to find the integral:
Step 1: Factorize the denominator.
The denominator x^2 + 6x + 13 cannot be factored using real numbers. However, we can complete the square to simplify it.
Completing the square:
x^2 + 6x + 13 = (x^2 + 6x + 9) + 4 = (x + 3)^2 + 4
Step 2: Rewrite the integrand using partial fraction decomposition.
We decompose the integrand 2x/(x^2 + 6x + 13) into partial fractions of the form A/(x - a) + B/(x - a)^2.
2x / [(x^2 + 6x + 13)] = A / [x + 3 + i2] + B / [x + 3 - i2]
Since the equation is equal for all x, we can create a common denominator:
2x = A(x + 3 - i2) + B(x + 3 + i2)
Expanding and gathering similar terms:
2x = (A + B)x + (3A + 3B) + (i2A - i2B)
Matching the coefficients of similar powers of x:
2x = (A + B)x
3A + 3B = 0
i2A - i2B = 0
From the second equation, we have A + B = 0, which implies A = -B. Substituting this into the third equation, we have i2A - i2B = 0, which simplifies to -2B - 2B = 0. Solving this equation, we find B = -i/2, and A = i/2.
Step 3: Rewrite the original integral using the partial fraction decomposition.
The integral of 2x/(x^2 + 6x + 13) can be written as:
∫[2x / (x^2 + 6x + 13)]dx = ∫[i/2(x + 3 - i2)]dx - ∫[i/2(x + 3 + i2)]dx
Simplifying the above equation:
∫i/2(x + 3 - i2)dx - ∫i/2(x + 3 + i2)dx
Step 4: Evaluate the integrals.
The integral of i/2(x + 3 - i2)dx is (i/2)ln|x + 3 - i2| + C1, where C1 is the constant of integration.
The integral of i/2(x + 3 + i2)dx is (i/2)ln|x + 3 + i2| + C2, where C2 is the constant of integration.
Step 5: Combine the results.
The final result is:
∫[2x/(x^2 + 6x + 13)]dx = (i/2)ln|x + 3 - i2| - (i/2)ln|x + 3 + i2| + C3
where C3 = C1 - C2 is the constant of integration.