a] Evaluate each of the following integrals il int _(0)^(pi )/(6)24sin^5Theta cosTheta dTheta (3 marks) ii] int _(0)^2(3x)/(sqrt ((2x^2)+1))dx (Taking positive roots only) (3 marks) b] Evaluate int _(0)^15xe^4xdx , correct to 3 significant figures (4 marks) c] Express (11-3x)/(x^2)+2x-3 in terms of partial fractions hence evaluate int (11-3x)/(x^2)+2x-3dx (5 marks) d] Determine the area enclosed between the curves y=x^2+1 and y=7-x

a] i) To evaluate the integral , we can use the substitution method. Let , then . The integral becomes . Integrating, we get . Therefore, the correct answer is .ii) To evaluate the integral , we can use the substitution method. Let , then . The integral becomes . Integrating, we get . Therefore, the correct answer is .b) To evaluate the integral , we can use integration by parts. Let and . Then and . Applying the integration by parts formula, we get . Integrating, we get . Therefore, the correct answer is .c) To express in terms of partial fractions, we can factor the denominator as . Then, we can write . Solving for and , we get and . Therefore, . To evaluate , we can integrate each term separately. The integral becomes . Integrating, we get , where is the constant of integration.d) To determine the area enclosed between the curves and , we need to find the points of intersection. Setting , we get . Solving this quadratic equation, we get and . Therefore, the area enclosed between the curves is . Integrating, we get . Therefore, the area enclosed between the curves is 11.