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The inverse Laplace transform of 1/x^(3/2) is 2/(sqrt(pi))*sqrt(x).

The Laplace transform of sin^2(t) is 2 / (s * (s^2 + 4)).

The Laplace transform of e^(t/2) is 1/(s - 1/2), provided s > 1/2.

The equation |a - 5| = |5 - a| is proven by considering two cases: when (a - 5) is greater than or equal to zero and when (a - 5) is less than zero. In both cases, the equation holds true, demonstrating that the absolute value of the difference between a and 5 is equal to the absolute value of the difference between 5 and a for all values of a.

The rational function 1/(3x^2 + 4x + 1) is decomposed into partial fractions as (3/2)/(3x + 1) - (1/2)/(x + 1) by factoring the denominator and solving for the unknown constants using strategic values of x.

The partial fraction decomposition of the rational expression x/((x+1)(x-4)) is (1/5)/(x+1) + (4/5)/(x-4).

The area of the region defined by the inequalities y ≥ x, y > -x + 1, and y < 10 is 90.25 square units.

The system of equations is solved by substitution, expanding, simplifying, using the quadratic formula to find x values, and then substituting back into the equations to find the corresponding y values. The solutions are (3, 4) and (-4, -3).

The solution to the system of equations x + 2y = 5 and 3x - y = 1 is x = 1 and y = 2, found using the elimination method.

The solution for x in the equation involving fractions with (x-1) and (x+1) in the denominator is x = √2 and x = -√2.