Final answer:
The integral of Ln(x+9) is calculated using integration by parts, leading to the result xLn(x+9) - x + 9Ln(x+9) + C, where C is the constant of integration.
Step-by-step explanation:
The integral of Ln(x+9) can be found using integration by parts. This method requires us to select parts of the integrand to be u and dv. For this integral, let u = Ln(x+9) and dv = dx. Now, we differentiate u to get du = (1/(x+9))dx, and we integrate dv to get v = x.
Applying the integration by parts formula ∫ u dv = uv - ∫ v du, we get:
Integral of Ln(x+9) dx = xLn(x+9) - ∫ (x/(x+9)) dx
To evaluate the remaining integral, we can perform a simple division which gives:
∫ (x/(x+9)) dx = ∫ (1 - 9/(x+9)) dx = x - 9Ln(x+9) + C
Therefore, the original integral can be written as:
xLn(x+9) - (x - 9Ln(x+9)) + C = xLn(x+9) - x + 9Ln(x+9) + C
Here, C represents the constant of integration.