Find the integral of:
dx/(x^2-4x)
To find the integral of dx/(x^2-4x), we can use a technique called partial fraction decomposition. Let's go step by step:
Step 1: Factor the denominator
We start by factoring the denominator x^2 - 4x. We can rewrite it as x(x-4).
Step 2: Decompose the fraction
Next, we need to decompose the fraction into partial fractions. We assume that the integral of the original fraction can be written as the sum of two fractions with simpler denominators:
dx/(x(x-4)) = A/x + B/(x-4)
Step 3: Find the values of A and B
To find the values of A and B, we clear the fractions by multiplying both sides of the equation by the common denominator, x(x-4):
1 = A(x-4) + Bx
Now we can simplify this equation and collect like terms:
1 = Ax - 4A + Bx
Comparing coefficients, we get two equations:
A + B = 0 (coefficient of x term)
-4A = 1 (constant term)
Solving these equations simultaneously, we find that A = -1/4 and B = 1/4.
Step 4: Rewrite the original fraction
Now that we know the values of A and B, we can rewrite the original fraction with the partial fractions:
dx/(x(x-4)) = -1/4 * 1/x + 1/4 * 1/(x-4)
Step 5: Integrate each partial fraction
Now we can integrate each partial fraction separately:
∫(-1/4) * 1/x dx + ∫(1/4) * 1/(x-4) dx
Simplifying:
-1/4 * ∫1/x dx + 1/4 * ∫1/(x-4) dx
Step 6: Evaluate the integrals
The integral of 1/x is ln|x|, and the integral of 1/(x-4) is ln|x-4|. Applying this to our equation:
-1/4 * ln|x| + 1/4 * ln|x-4|
And there you have it! The integral of dx/(x^2-4x) is:
-1/4 * ln|x| + 1/4 * ln|x-4| + C
Note: The "+ C" represents the constant of integration.