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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 eigenvalues of the matrix [[3, 1], [1, 3]] are 4 and 2.

The eigenvalues of the matrix [[0, 1], [-1, 0]] are i and -i.

The eigenvalues of the given 3x3 matrix are 1, 2, and 2.

The problem involves finding the equation of a circle given a point on the circle, the radius, and the fact that the center of the circle lies on a given line. Two possible circles satisfy the conditions, each with a different center, and the equations for both circles are derived.

The solution finds the trace lines of a plane by setting one coordinate to zero and solving for the other two coordinates in terms of the remaining parameter. This results in parametric equations for the trace lines in the xy, xz, and yz planes.

The equation represents an ellipse centered at (-2, 0) with a vertical major axis of length 4 and a horizontal minor axis of length 2.

The solution to the equation \sqrt{(X - 4)(X + 4)} = 3 is X = 5 and X = -5.

The integrals were evaluated using trigonometric substitution, partial fraction decomposition, and substitution. Integral c) was solved using trigonometric substitution, resulting in x/(√(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 solution involves finding the values of a and b, given a*a = 8 and a*b = 24. Then, b*b is calculated, resulting in a final answer of 72.

Cramer's rule is used to solve a system of four linear equations with four unknowns, resulting in the values of x1, x2, x3, and x4.

The equation of the line equidistant from the parallel lines y = (1/3)x - 4 and y = (1/3)x + 1 is y = (1/3)x - 5/3.

The solution finds the equation of a line g' that is the reflection of line g across line s. It involves finding the intersection point of g and s, projecting a direction vector of g onto a normal vector of s, and then calculating the direction vector of g'. Finally, the equation of g' is determined using the intersection point and the direction vector of g'.

The square root of 19 squared plus 19 plus 20 is calculated to be 20.

To find (a+b)^2, given a^2 + b^2 = 29 and ab = 10, substitute the given values into the equation (a+b)^2 = a^2 + 2ab + b^2, resulting in (a+b)^2 = 49.

The expression 3³ ÷ 3 ÷ 3 simplifies to 1.

The only real solution to the cubic equation a^3 + a^2 - 36 = 0 is a = 3.

The solution to the equation X^3 - X^0 = 26 is X = 3.

24 is 60 percent of 40.

The determinant of the given 4x4 matrix is calculated using cofactor expansion along the fourth row, resulting in a determinant of -21.

The determinant of the 2x2 matrix [[2, 2], [3, 5]] is calculated as (2 * 5) - (2 * 3) = 4.

The equation (x-1)^(|x|-4) = 1 has solutions x = -4, 0, 2, and 4.

The difference of squares formula is used to evaluate the expression 100^2 - 99^2, resulting in 199.

To find the smallest possible values for A and B in a 10-digit number divisible by both 8 and 9, the last three digits (9B8) must be divisible by 8, and the sum of all digits must be divisible by 9. This leads to B=2 and A=3 as the smallest possible values.

The equation x^2 - 2x = 0 has two solutions, x = 0 and x = 2, not just x = 2.

The solution for b in the equation 2^(b^2+2) = 64 is b = 2.

To find 2^(m+n), solve for m and n in the equations 2^m = 1 and 2^n = 16, then substitute the values into the expression.

The expression 0^0 is considered an indeterminate form in calculus because different approaches to evaluating it yield different potential values. While some approaches suggest a value of 1 and others suggest 0, the limit does not have a single, well-defined value. In some contexts, like combinatorics, it is defined as 1 for convenience.

The magnitude of the given row vector is calculated, resulting in 3.