182Ech5S - Math 182 Chapter 5 Exam Graphing and symbolic...

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Math 182 Chapter 5 Exam Graphing and symbolic algebra calculators are prohibited on the following exam. All necessary calculations must be shown. Unjustified answers will receive very little credit. With the exception of the numerical approximations section, all answers must be exact. PROBLEM POSSIBLE EARNED Substitution 5 Integration by Parts 20 Trig Substitution 20 Partial Fractions 15 Numerical Approximations 20 Improper Integrals 20 TOTAL 100 Extra Credit 5 Substitution Calculate the following: integraldisplay e π e π 4 sin(ln x ) x dx We will let u = ln x so that du = dx x . Then when x = e π 4 we have that u = ln( e π 4 ) = π 4 and similarly when x = e π we have that u = ln( e π ) = π . This gives us = integraldisplay π π 4 sin u du = - cos u | u = π u = π 4 = cos u | u = π 4 u = π = cos( π 4 ) - cos( π ) = 1 2 + 1 Integration by Parts Calculate the following: integraldisplay x 3 e x 2 dx We first make the substitution w = x 2 so that dw = 2 xdx and hence xdx = 1 2 dw . This gives us = 1 2 integraldisplay we w dw Recall the parts formula integraltext udv = uv - integraltext vdu . We can let u = w so that du = dw and dv = e w dw so v = e w . Integration by parts then gives = 1 2 bracketleftbigg we w - integraldisplay e w dw bracketrightbigg = we w - e w 2 + C = x 2 e x 2 - e x 2 2 + C
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Trig Substitution Consider integraldisplay x 5 (1 + x 2 ) 4 dx If you wish to simply evaluate the above integral you may do so with your own carefully explained calculations. However, the following steps are the best guide to partial credit if you cannot finish the problem completely. (a) By noting that (1 + x 2 ) 4 = 1 + x 2 8 introduce a right triangle with angle θ and sides labeled appropriately to make a trig substitution.
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