W rite the full worked out solutions for Examples 5, 6 and 7Comple te Exercise to Try 2. Sh ow all work

Example 5 Evaluate [ to sin(f) dt A final example shows how we can solve an indefinite integral by exploiting the cyclic properties of trig derivatives in combination with integration by parts. Example 6 Evaluate fe" sin(x) dr Evaluating Definite Integrals Using Integration by Parts Here's the formula for evaluating definite integrals using Integration by Parts: (6) Example 7 Evaluate ._"(x + 2) sin(x) dx and y = lim lim E f(1,)Ar (10) J. f(z) de Equations () and (10) are the formulas for the coordinates of the centroid C' of R. Activity To Try 1 Click here to navigate to a graph of the region R, which displays the region between the graphs of f(x) = x and g(x) = x2 on the interval [0, 1). The area of R is approximated with n = 10 evenly spaced rectangles. Use the graph the answer the questions below. (a) What is the height dimension of the 4th rectangle? (b) What is the height dimension fo the 5th rectangle? (c) Using f(z) and g(z), write a formula for the height dimension of the ith rectangle R,. Next, click here to simulate the limit as n -+ Do. Use the slider to increase n to 100. What values do you think the coordinates of the centroid are getting arbitrarily close to Record your answer below. Exercise To Try 2 Use formulas (@) and (10) to find the exact centroid of the region R bounded by the curves f(x) = x and g(x) = x on the interval [0, 1]