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Published on October 26, 2024
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).
xยฒ + yยฒ = 25 y = x + 1
The function f(x,y) = -2x^2 + 3xy^2 - 3y^2 + 4 has a local maximum value of 4 at (0,0).
The double integral, using polar coordinates and integration by parts, evaluates to ฯ(e^2 + 1).
The solution attempts to derive a reduction formula for the integral \(\int_0^\infty \frac{x}{\cosh^n(x)} dx\) using integration by parts and Feynman's trick, but ultimately finds that expressing the integral in terms of hypergeometric functions is a more suitable approach.
The integral \int_{0}^{\infty} \frac{x}{(\cosh x)^2} dx evaluates to ln(2).
To evaluate the limit, direct substitution results in an indeterminate form (0/0). Factoring the numerator reveals a common factor of (x-3), which cancels with the denominator. Substituting x=3 into the simplified expression yields a final answer of 58.
The integrals were evaluated using trigonometric substitution, partial fraction decomposition, and substitution. Integral c) was solved using trigonometric substitution, resulting in x/(โ(1-x^2)) + C. Integral d) diverged to infinity. Integral e) was solved using partial fraction decomposition, leading to a solution involving natural logarithms and a constant term.
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.