Use trigonometric substitution to verify that

∫0xa2−t2√dt
〖√(a^2-t^2 ) dt〗=1/2 a^2 sin^(-1)⁡〖x/a〗+1/2 x√(a^2-x^2 )

use the figure to give trigonometric interpretations of both terms on the right side of the equation in part (a).

Solve for x and y: 217x + 131y = 913; 131x + 217y = 827

Prove (a) cosh2(x) − sinh2(x) = 1 and (b) 1 − tanh 2(x) = sech 2(x). solution (a) cosh2(x) − sinh2(x) = ex + e−x 2 2 − 2 = e2x + 2 + e−2x 4 − = 4 = . (b) we start with the identity proved in part (a): cosh2(x) − sinh2(x) = 1. if we divide both sides by cosh2(x), we get 1 − sinh2(x) cosh2(x) = 1 or 1 − tanh 2(x) = .

Atourist boat is used for sightseeing in a nearby river. the boat travels 2.4 miles downstream and in the same amount of time, it travels 1.8 miles upstream. if the boat travels at an average speed of 21 miles per hour in the still water, find the current of the river.