a) Which integration method would be used to evaluate 3x /(x1)(x+2)d Answer: c b) The general antiderivative can be written in the form 3x =A1 P Bl /(x1)(x+2)dx n|x |+ n|x Q|+C, where the coefficients A, B, P and Q are integers, and C is a constant of integration. Find this antiderivative by entering numerals for A, P, B and Q in the blanks below to make a true statement. Note the pattern of the signs in the expression, and enter your coefficients accordingly. f (x_13)(x+2)d 1n |x |+ 1n |x I + C The values in the blanks go in order (left to right): A, then P, then B, then Q. Consider the region bounded by the graphs of x = (y - 1) and y = x-1. (4, 3) 3 x = (y - 1)2 2 1 (1, 0) 0 2 3 y =x-1 i) Find a definite integral that represents the area of the region. a) lo (y + 1) - (y - 1)2 dy b) So ( Vx + 1 ) - (x - 1) dx c) lo Vy - 1 - (v+ 1)dy d) 18 ( y - 1)2 - (1 -1) dy e ) fi ( x - 1) 2 - Vx - Idx Answer: (Select one of the options from the pull-downi) Find a definite integral that represents the area of the region. a) fo(y + 1) - (y-1)2 dy b) So ( Vx + 1) - (x - 1) dx c) lo Vy - 1 - (y+ 1) dy d) Jo ( y - 1)2 - (y - 1) dy e ) fi ( x - 1)2 - Vx - Idx Answer: (Select one of the options from the pull-down menu) ii) Calculate the area of the region. You may use a numerical integration routine. If the answer is not an integer, enter it in decimal form accurate to at least one decimal place. Calculators: TI-84 Desmos Simpson's Rule Approximator Area = (Enter a decimal number)In this problem, we will outline the process of evaluating the integral by trigonometric substitution. For each part of this problem, you will drag and drop item(s) into their correct locations in the image below. a) What is the correct trigonometric substitution to use for this integral? Drag one of the items from the first group (white background) into the box labeled "Trigonometric substitutionz". b) Based on your trigonometric substitution, label the sides of the right triangle with the appropriate expressions from the second group of items (blue background). Drag each expression to its corresponding side in the right triangle diagram, in the area labeled "Right triangle:". The angle 9 and the right angle are marked in the figure. c) After performing the correct trigonometric substitution and simplifying, the given integral is transformed into the following form. (You may accept this part as given). f 3 sin2(6) d6 \fX = 9tan(0) X = 9sin(0) X = 9sec(0) V(81-x2 ) 9 V (x2+81) V (x2-81) X V81-x2 + C 2187x xV81-x2 NIW arcsin + C 54 1 arcsec + C 27 XVx2-81 NIW In x + Vx2 - 81 + C 54 1 9+V81-x2 In + C 243 X Suppose that f is a differentiable function with derivative f' . Some values of f and f' are listed in the table below. Based on the givenSuppose that f is a differentiable function with derivative f. Some values of f and f are listed in the table below. Based on the given information, find the exact value of each of the following definite integrals. Important: Enter non-integer values as decimals. 0 1 4 f(2) 2 5 0 11 f' (a) 3 9 4 2 a) / f' (sin (x)) cos(x) dx = (Enter a number) 4 b ) 2f" ( x ) dx = (Enter a number) f' (x ) C ) dx = ( f ( x ) ) (Enter a number)Given: 42 f(x) dx = 12 11 42 g(x) dx = 7 11 42 h(x) dx = 3 11 Find: The numerical value of each definite integral below. . 42 a ) (2f ( x) - 38( x) + 5h(x) dx = (Enter a numeral) 11 11 b) f(x) dx = (Enter a numeral) 42