16 SOLVE THE PROBLEM AND PLEASE SHOW YOUR WORK Evaluate cos7

16. SOLVE THE PROBLEM AND PLEASE SHOW YOUR WORK!

Evaluate cos7x dx The blue of a solid the region between the curve y = sin x / 2 and the interval [0,2 pi] on the x-axis. The cross sections perpendicular to the x-axis are semicircles with bases running from the x-axis to the curve. Find the volume of the solid. Round your answer to three decimal places

Solution

first of all dividing the expression by long division method (20x^2 + 28x)/(10x-1) = 2x + 30x/(10x-1) I = [2x + 30x/(10x-1)]dx I = [2x + (30x-3 + 3)/(10x-1)]dx I = [2x + 3 + 3/(10x-1)]dx integrating wrt x I = x^2 + 3x + 3/10 ln|10x-1| + C