Final Answer:
The integral ∫(1/(x² - 3))e⁽⁻ˣ⁾ dx from 0 is equal to -0.5e³⁄² + 0.5e⁻³⁄².
Step-by-step explanation:
To evaluate the given integral, we can use partial fraction decomposition. First, factorize the denominator x² - 3 into (x + √3)(x - √3). The partial fraction decomposition is then A/(x + √3) + B/(x - √3). Multiply both sides by the common denominator (x + √3)(x - √3) to clear fractions. After simplifying and solving for A and B, the decomposition becomes A(x - √3) + B(x + √3) = 1. Now, integrate each term separately.
The integral of A(x - √3) with respect to x is A⁄2 * (x² - 2√3x), and the integral of B(x + √3) is B⁄2 * (x² + 2√3x). Evaluate these integrals from 0 to infinity, subtracting the result at 0 from the result at infinity. After plugging in the values, the final answer simplifies to -0.5e³⁄² + 0.5e⁻³⁄².
In summary, by applying partial fraction decomposition and integrating each term separately, we find the definite integral evaluates to -0.5e³⁄² + 0.5e⁻³⁄². This result represents the area under the curve of the given function from 0 to infinity.