please calculate me 1, 2, 3, 4, 5. Please write clearly for me. Thanks.

1 Substitution in indefinite integrals Theorem (The Substitution Rule). Let u = g(x) be a differentiable function whose range is I. Let f be a continuous function on I. Then J f(g(z) . 9(2)de = [ fu)du. 2 Substitution in definite integrals Theorem (The Substitution Rule for definite integrals). Let u = g(x) have a continuous derivative on [a, b]. Let f be a continuous on the range of 9. Then [s(g(2) . "(2)da = fu)du. N Mango Homework PDF-beta 1. Compute the integral (a) f sin(ma) da by letting u = Tx. (b) fer/2dx by letting u = x/2. Show that you got the correct answer by differentiating the computed function. 2. Compute the integral and then verify that you got the correct answer by differentiation. (a) faz ja3 + 7dx rina 3. Evaluate the definite integral (a) 2 V7 - 2xdx [Note originally there was a different limit of intergation - we will accept solutions with the original limit too] ( b) So te -12/2dt. 4. (a) Find the average value fna,bj Of fn(x) = sin(nx) on the interval [a, b]. (b) Based on a sample graph of fn(x) what do you expect about the tendency of the averages: fila,bj, f2[a,b)> ...? Verify your expection by formally computing lim fn[a,b]. 1-+0o 5. The arc of the parabola y = x2 from (3,9) to (4, 16) is rotated about the y-axis. Find the area of the resulting surface. Please use Woframalpha only to check your final answer. Show all steps in solving the appropriate integral. Learning System . mangeroot 2008-2021