du u C sec( u) tan u du secu C e du e C a u a a Basic Integration Trigonometric Functions: du ln u u Helpful to Know: Inverse Trigonometric
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1 Student Study Session Eam Solutions We have intentionally included more material than can e covered in most Student Study Sessions to account for groups that are ale to answer the questions at a faster rate. Use your own judgment, ased on the group of students, to determine the order and selection of questions to work in the session. Be sure to include a variety of types of questions (multiple choice, free response, calculator, and non-calculator) in the time allotted. Basic Integration kf( udu ) k f( udu ) [ f ( u) gu ( )] du fudu ( ) gudu ( ) du u C n n u u du C n du ln u C u u u a du a C ln a u u e du e C Inverse Trigonometric du arcsin u C a u a Trigonometric Functions: sin( udu ) cosuc cos( udu ) sin u C sec ( ) csc ( ) udutanuc uducot uc sec( u) tan u dusecuc csc( u)cot u ducscuc Helpful to Know: tan( u) du ln cos C cot( u) du ln sin u C du u arctan C a u a a
2 Students approach questions requiring the use of u-sustitution or parts in a variety of ways. The solutions to questions and show alternative approaches to each type of question.. B (97 AB/BC) ln (ln ) (ln ) ln d d. C (969 AB8) d e ( ) d e C C e e. B ( AB8/BC8) cos d cos d sin( ) c. A (969 AB9) cos d ln(sin ) lnln ln ln ln sin Alternatively, rewrite the entire integral in terms of u. u sin When, u sin ; when du cos d, u sin. u cos u d d du u u u u cos ln( ) lnln ln ln ln sin sin Alternatively, integrate in terms of u; replace efore evaluating the definite integral. u sin du cos d cos d cos d du ln( u ) ln(sin ) lnln ln ln ln sin sin u Copyright National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
3 5. B (998 BC8 appropriate for AB). Let sin cos d sin cos d cos C y() (cos)., cos C, so C. 6. A (985 BC appropriate for AB) u, du d; when, u and when, u u u d du du u u 7. B (97 AB8/BC8) Let z c, so 5 f( c) d c f( z) dz. c 8. C (997 AB) ( f ( ) 5) d f ( ) d 5 d a 5 a 5( a ) 7 a a a. a a 9. E (998 AB8). Since F is an antiderivative of f, f( ) d F( ) F(6) F(). A (988 BC6) u vd e + e - e e d e e C Alternatively, using a non-taular method with parts, u dv e d du d v e e d e e d e e C Copyright National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
4 . E (99 BC9) u vd sec + sin tan - cos ln cos sec d tan ln cos C. B (969 BC) u dv cos du d v sin cos d sin sin d. cos d f( ) sin d sin sin d f( ) sin d (using sustitution) sin C f( ).. B (985 BC ) u f ( ) dv sin d du f ( ) d v cos f ( ) sin( ) d f( )cos cos f ( ) d f( ) sin( ) d f( )cos cos d f ( )coscos f( ) df( )cos cos d (using sustitution) (), so f f( ) c.. E (8 BC) u f ( ) v g( ) du f ( ) d dv g( ) d f( gd ) ( ) f( g ) ( ) gf ( ) ( d ) f ( ) g ( ) d f () g () f () g () 5 (using sustitution) f( ) g( ) d() ( ) 5 5 Copyright National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
5 5. A (985 BC) A ( ) B ( ) Let : B( ), sob ; Let : A, soa. d ln ln c ln C 6. A (8 BC9) 7 A( ) B( ) 7 Let : A 7 ; Let. : B. ( ) d ln ln C 7. E (97 BC6) ( ) ( ) d lim d lim ln lim ln ln so this limit diverges. 8. A (988 BC7) lim d lim lim. 9. E (99 BC) lim 9 dlim (9 ) lim 9 9. This limit diverges.. D (997 BC89) Using a calculator, () () f f d Copyright National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
6 . B BC5d (d) gd ( ) lim d lim( ) This limit is not finite, so the integral is divergent. gd ( ) d lim : gd ( ) : indicates integral diverges : gd ( ) : finite limit as Copyright National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
7 . 97 BC5 Let I e d Let u dv e d du d v e I e e d e e lim e d e d lime e lim e e lim e e No points shown. Integral converges to.. 8 BC5c (c) : f () f ( ) d 7 ( ) e d u dv e d du d v e f () 7 ( ) e e d 7e e : usesinitial condition :integrationy parts : answer Copyright National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
du u C sec( u) tan u du secu C e du e C a u a a Trigonometric Functions: Basic Integration du ln u u Helpful to Know:
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