Evaluate the integral! 17 sin ?() cos 3 ( # ) ox Step 1 since 17 sin? (x) cos (x) dx has an odd power of of cos(x), we will convert all but one power to sines. We know that cos ( x) - 1 - 1 - sin = ( x ) . Step 2 Making this substitution using 17 sin ? (x ) cos ( x ) dx gives us 17 sin2(x) (1 - sin2(x)) cos(x) ax = / 17 sin?(x) cos(x) ex - / (17 sint (x)co=() 17 sin' (z) cos (z) ) dx. Step 3 Since cos(x) is the derivative of sin (x). then /|is sinic cosco) ox can be done by substituting u - sin (5) sin ( x) and du = cos (x cos (x) Step 4 With the substitution u = sin(x), we get 17 sin 2 ( x ) cos ( x ) dx = 17 / 42 du . which integrates to 17 3 -17 5 + C. X Substituting back in to get the answer in terms of sin(x), we have 17 sin? (x) cos(x) dx - 17 sin' (x) - sins (x)| + C . X