Final answer:
To evaluate the integral, use the partial fraction decomposition method and integrate term by term to obtain the solution.
Step-by-step explanation:
To evaluate the integral ∫[infinity]∫⁰ 12dx/(x²+144), we can first note that the integrand is a rational function with a denominator of x²+144. The denominator is always positive, so the behavior of the integrand is determined by the numerator. Since the numerator is a constant (12), the integrand approaches 0 as x approaches infinity. Therefore, the integral converges.
To evaluate the integral, we can use the partial fractions method. The denominator x²+144 can be factored as (x+12)(x-12). By decomposing the integrand into partial fractions, we get 12/((x+12)(x-12)).
If we let A/(x+12) + B/(x-12) be the partial fraction decomposition, we can find the values of A and B by equating the numerators of both sides. We get A(x-12) + B(x+12) = 12. By expanding and equating coefficients, we find A = B = 1/24.
Substituting the values of A and B back into the partial fraction decomposition, we have 1/24(x+12) - 1/24(x-12) as the integrand. Now we can integrate term by term, and we get (1/24)ln|x+12| - (1/24)ln|x-12| + C, where C is the constant of integration.