Evaluate the integral using the indicated trigonometric substitution. (Use C for the constant of integration.) Z _ / V 4X 25 dX, X = i sec(0) . X 2 Sketch and label the associated right triangle. CHOICE A CHOICE B V 4 x2 25 2 x ( A 2 X A 4 X2 25 I I 5 5 0 0 CHOICE C CHOICE D \fEvaluate the integral. (Use C for the constant of integration.) +2 3 + 2x - x2 dx Need Help? Read ItA water storage tank has the shape of a cylinder with diameter 10 feet. It is mounted so that the circular cross-sections are vertical. If the depth of the water is 7 feet, what percentage of the total capacity is being used? (Round your answer to one decimal place.) -x % Need Help? and" (a) Use trigonometric substitution to verify that a2 - +2 dt = _a2 sin-1 X + -XV a2 - x2. Since the integral Va2 - +2 dt involves an expression similar to va2 - x2, we will use the substitution t = a sin(0). With this substitution, dt = a cos(0) do. Identify the operations, theorems, and/or identities that justify each step in the derivation below. Va2 - +2 dt = sin (x/ 2)a cos(0)(a cos() de) by substitution t = a sin(0) Jo = a2 / sin "(x / a) cos2(0) de ---Select--- 32 sin -1(x / a) = (1 + cos(20)) de ---Select--- 2 Jo a2 sin - 1 (x / a ) = ---Select--- 2 0 + _ sin ( 20 ) Jo a2 0 + sin(0) cos(0) sin (x / a) = 2 ---Select--- o = sin - 1 X + X 0 ---Select--- v N a = a sin -1 N K 2 + xVaz - x2 by simplificationy t X (D X . , The integral / \\/52 _ t2 dt represents the area under the curve y = x/az _ t2 between the vertical lines t = 0 and t = . The gure shows that 0 this area consists of a triangular region and a sector of the circle a2 = area is . The sector has area aza = 32 sin_1( Need Help? mm . The triangular region has base x and \\/ a2 x2, 50 its