MATH 181 Study Guide - Final Guide: Trigonometric Substitution, Random Variable
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Solution: we begin by using the u-substitution method. Then du = 2x dx and we get: Z 2x 1 x4 dx = z 2xq1 (x2)2 dx. We now use a trigonometric substitution to evaluate this integral. Then du = cos d and we get: Z 1 u2 du = z p1 sin2 (cos d ) 2 sin(2 ) + c sin cos + c. To write the result in terms of u we use the fact that: to get: = arcsin u, sin = u, 2 arcsin u + cos = 1 u2 u 1 u2 + c. Finally, we write the answer in terms of x replacing u with x2: Problem 2 solution: determine whether the following integrals converge or not: Solution: the rst integral is improper due to the in nite upper limit of integration. We will evaluate the integral by turning it into a limit calculation. We use the u-substitution method to compute the integral.
