Evaluate using long division.
The integral of x/6x-5 dx
How do I divide this?
To evaluate the integral ∫(x/6x-5)dx, you can use long division. Here's how to do it step by step:
Step 1: Write the integral in a split form, separating the numerator and the denominator:
∫(x/6x-5)dx = ∫(x/[6x - 5]) dx
Step 2: Divide the denominator by the numerator using long division:
x
6x - 5 | x
- (Divide x by 6x)
6x
- (Multiply 6x by -5 and subtract from 6x - 5)
5
Step 3: Rewrite the integral as a sum of a quotient and a remainder:
∫(x/[6x - 5]) dx = ∫(1 + (5/[6x - 5])) dx
Step 4: Evaluate the integral of the quotient and the remainder separately:
∫dx + ∫(5/[6x - 5]) dx
Step 5: Integrate each term:
The integral of dx is x.
The integral of (5/[6x - 5]) dx is (1/6)ln|6x - 5|.
Step 6: Combine the results:
∫(x/6x-5)dx = x + (1/6)ln|6x - 5| + C
Therefore, the evaluation of the integral is x + (1/6)ln|6x - 5| + C, where C is the constant of integration.