MAT137Y1 Study Guide - Midterm Guide: Trigonometric Substitution, Conditional Convergence, Inverse Trigonometric Functions

• School: UTSG
• Department: Mathematics
• Course: MAT137Y1
• Professor: All

Document Summary

2: [4 points] compute the following integral. We can integrate by decomposing the fraction: (cid:90) 3 dx x2 + x. 2 (ln(x + 1))(cid:12)(cid:12)3 x + 1 dx. = ln(3) ln(2) (ln(4) ln(3)) 1: [5 points] compute the following integral (cid:90) Note: your nal answer should be written without using any trigonometric or inverse trigonometric functions. We perform the trigonometric substitution tan( ) = x, sec2( )d = dx. (cid:90) (cid:0) . 1 + x2(cid:1)5 = dx (cid:17)5 sec2( ) d (cid:90) (cid:16)(cid:112)1 + tan2( ) (cid:90) sec2( ) d (cid:90) (cid:90) (cid:0)1 sin2( )(cid:1) cos( ) d cos3( ) d sec5( ) In the second last line we used the fact that sin( ) = x x2 + 1 provided that tan( ) = x. 2: [9 points] determine whether each of the following series is absolutely conver- gent (ac), conditionally convergent (cc), or divergent (d). Note: you won"t get any points without a correct justi cation.