prob3practicesol - Math 151 Spring 2008 Problem Set 3 Integration by Parts Solutions of the Practice Problems Remark An arbitrary constant can be added

# prob3practicesol - Math 151 Spring 2008 Problem Set 3...

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Math 151 Spring 2008 Problem Set # 3 Integration by Parts Solutions of the Practice Problems Remark: An arbitrary constant can be added to an antiderivative. 2. Set u = x 2 and dv = e x/ 3 dx so that du = 2 xdx and v = Z e x/ 3 dx = 3 e x/ 3 . Therefore, Z x 2 e x/ 3 dx = Z udv = uv Z vdu = x 2 ³ 3 e x/ 3 ´ + 3 Z e x/ 3 2 xdx = 3 x 2 e x/ 3 + 6 Z xe x/ 3 dx. We set u = x and dv = e x/ 3 dx so that du = dx and v = 3 e x/ 3 . Therefore, Z xe x/ 3 = Z udv = uv Z vdu = x ³ 3 e x/ 3 ´ + 3 Z e x/ 3 dx = 3 xe x/ 3 9 e x/ 3 . Thus, Z x 2 e x/ 3 dx = 3 x 2 e x/ 3 + 6 Z xe x/ 3 dx = 3 x 2 e x/ 3 + 6 ³ 3 xe x/ 3 9 e x/ 3 ´ = 3 x 2 e x/ 3 18 xe x/ 3 54 e x/ 3 (a constant can be added). 3. Set u = x and dv = cos ( x/ 2) so that du = dx and v = Z cos ( x/ 2) dx = 2 sin ( x/ 2) . Therefore, Z x cos μ 1 2 x dx = Z udv = uv Z vdu = x (2 sin ( x/ 2)) 2 Z sin ( x/ 2) dx = 2 x sin ( x/ 2) + 4 cos ( x/ 2) . 1
6. We set u = x 2 and dv = cosh ( x ) dx so that du = 2 xdx and v = Z cosh ( x ) dx = sinh ( x ) . Thus. Z x 2 cosh ( x ) dx = Z udv = uv Z vdu = x 2 sinh ( x ) 2 Z x sinh ( x ) dx. As in Problem 5, Z x sinh ( x ) dx = x cosh ( x ) sinh ( x ) . Therefore, Z x 2 cosh ( x ) dx = x 2 sinh ( x ) 2 Z x sinh ( x ) dx = x 2 sinh ( x ) 2 ( x cosh ( x ) sinh ( x )) = x 2 sinh ( x ) 2 x cosh ( x ) + 2 sinh ( x ) .