Math Archive: Questions from 2024-02-5

A line y = 2x - 1 cuts an oval (x/2)^2 + y^2 - 1 = 0 into two regions. Which of the following represents the area of the smaller region enclosed between the given line and the given oval. Select one: a. 15/17 -1 p 4 - 4y 2 - 1+y 2 dy b. 16/17 0 1 - x 2 2 - 2xdx c. R 15/17 -1 x 2 2 + y 2 - 2x + 1dx d. R 16/17 0 x 2 2 + y 2 - 1 – 2xdx

Given x = 4 sec(θ), express ln |θ| + cot(θ) + C in terms of x and simplify as much as possible. Select one: a. ln |θ| + cot(θ) + C = ln |arccot x 4| + √ 4 x2+16 + C b. ln |θ| + cot(θ)+C = ln | arcsec(4x)|+ √ 4 16x2-1 +C c. ln |θ|+cot(θ)+C = ln | arccot(4x)|+ √ 4 16x2+1 + C d. ln |θ| + cot(θ) + C = ln |arcsec x 4 | + √ 4 x2-16 + C

Evaluate ∫ cosn sin2k+1(x)dx where k is an integer. Consider all steps of your calculations. Select one: a. n must be negative. b. n must be positive. c. n must be odd. d. n must be even. e. n can be any real number.