To solve this integral, we will need to work through it step by step. Given the expression you provided:
∫[π(3ln(3)^2 - 6ln(3) + 6 - ln(1)^2 + 2ln(1) - 2)]
Let's simplify and rewrite it before integrating:
∫[π(3ln(3)^2 - ln(1)^2 - 6ln(3) + 2ln(1) + 4)]
Now, since ln(1) equals 0, we can simplify further:
∫[π(3ln(3)^2 - 6ln(3) + 4)]
To integrate this expression, we will use the power rule for integrals. The general form of the power rule is:
∫[x^n] dx = (1/(n+1)) * x^(n+1) + C
Applying the power rule to our integral:
∫[3ln(3)^2 - 6ln(3) + 4] = (π/3) * ln(3)^3 - (π/2) * ln(3)^2 + 4π * x + C
So, the exact value of the integral is:
(Ï€/3) * ln(3)^3 - (Ï€/2) * ln(3)^2 + 4Ï€ * x + C
Please note that "x" is the variable of integration, so if you wanted a definite integral within a specific range, you would need to substitute the upper and lower limits of integration into the expression and evaluate it accordingly.