Math 2924 Improper Integrals - Arkansas Tech University MATH 2924 Calculus II Dr Marcel B Finan Solutions to Assignment 6.6 Exercise 1(a This is an

# Math 2924 Improper Integrals - Arkansas Tech University...

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Arkansas Tech University MATH 2924: Calculus II Dr. Marcel B. Finan Solutions to Assignment 6.6 Exercise 1 (a) This is an improper integral of type 2 where the integrand is discontin- uous at x = 1 . (b) This is an improper integral of type 1 where the interval of integration is unbounded. (c) This is an improper integral of type 1 where the interval of integration is unbounded. (d) This is an improper integral of type 2 where the integrand is discontin- uous at x = 0 Exercise 5 First, letting u = x - 2 , we find Z dx ( x - 2) 3 2 = Z u - 3 2 du = - 2 u = - 2 x - 2 + C. Thus, Z 3 dx ( x - 2) 3 2 = lim b →∞ 2 - 2 b - 2 = 2 so that the improper integral is convergent Exercise 7 First, letting u = 3 - 4 x, we find Z dx 3 - 4 x = - 1 4 Z du u = - 1 4 ln | u | = - 1 4 ln | 3 - 4 x | + C. Thus, Z 0 -∞ dx 3 - 4 x = lim a →-∞ - ln 3 4 + 1 4 ln | 3 - 4 a | = so that the improper integral is divvergent 1
Exercise 13 First, letting u = x 2 , we find Z xe - x 2 dx = 1 2 Z e - u du = - 1 2 e - u = - 1 2 e - x 2 + C.