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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1 Integration Techniques for AB Eam Solutions We have intentionally included more material than can be covered in most Student Study Sessions to account for groups that are able to answer the questions at a faster rate. Use your own judgment, based 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 u arcsin C a u a Trigonometric Functions: sin( udu ) cos uc 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 Integration Techniques for the AB eam Students approach u-substitutions questions in a variety of ways. The solution to question demonstrates three different approaches. Multiple Choice Questions Solutions. E (97 AB7) d d ( ) ( ) ( ) Alternatively, rewrite entire intregral in terms of u u when du d du d d ( ) ( ) d, u and when u u u du u u u, u ( ) Alternatively, integrate in terms of u; replace before evaluating definite integral u du d du d d ( ) ( ) d ( u) du u ( ). B (985 BC appropriate for AB) () d d ln (ln 8 ln ) Copyright National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

3 . B (97 AB/BC) Integration Techniques for the AB eam ln (ln ) (ln ) ln d d. E (998 AB7/BC7) e e e d = ln d = e e 5. C (997 BC appropriate for AB) 5 6 ( ) d d 5 5 Copyright National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at 6. C (969 AB8) d e ( ) d e C C e e 7. B (8 AB) sin( ) d cos( ) d sin( ) d cos( ) d cos( ) sin( ) C 8. B ( AB8/BC8) cos d cos d sin( ) C 9. A (969 AB9) cos d ln(sin ) lnln ln ln sin. B (998 BC8 appropriate for AB) sin cos d sin cos d cos C Let, cos C, so C. y() (cos)... A (985 BC appropriate for AB) u, du d; when, u and when, u u u d du du u u

4 . C (988 AB) k 8 k k k 7, k Integration Techniques for the AB eam. B (97 AB8/BC8) Let z c, so 5 f ( c) d c f ( z) dz. c. C (997 AB) b ( f ( b b b ) 5) d f ( ) d 5 d a b 5 a b 5( b a ) 7 b a a a a 5. A (998 AB/BC) Since f is linear, its second derivative is zero. The integral gives the area of a rectangle with zero height and width b a. This area is zero. 6. A (998 AB) k k d ( k ( ) ( k 7) only when k. 7. E (998 AB8) Since F is an antiderivative of f, f( ) d F( ) F(6) F() a. 8. D (997 AB9) I. f ( ) (sin )(cos ) sincos II. f ( ) (cos )( sin ) sin cos III. f( ) ( sin( ))() sin( ) (sin cos ) sin cos The derivative of the first and last functions are equal to f ( ). 9. C (8 BC86 appropriate for AB) dy d and y when. y C; C, so C ; therefore, f () 8 y and. C (998 AB88) Using a calculator, (ln ) F(9) F() t dt 5.87 t 9. Copyright National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

5 Free Response Solutions. 99 AB5 if (a) f if f if Integration Techniques for the AB eam : if : if : (b) C if f C if f CC f C C if f if f : Antiderivative for with or : Antiderivative for without C s : Constants : Antidifferentiates absolute value function If incorrect on one branch : Constant of if not a split function or absolute value function NOTE: No penalty for domain error in Part (b) if consistent with domain in Part (a) (c) Consistency with given f ' : Increasing and continuous on,. : Concave up on, and concave down on,. : f, lim f, lim f or if f not used in Part (b), then lim f lim f. If student goes outside domain, loses rd point. Copyright National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

6 . 989 BC (appropriate for AB) f 8 c (a) f c f d d f 5 : f Integration Techniques for the AB eam with c : Sets f : : Finds c : uses his/her f to find f with constant : uses, to find d (b) d : integrand other than f in part a : Set up : Incorrect b a : Incorrect limits : antidifferentiation and evaluation. 988 AB6 ab 6 f a b ab 8 a, b 6 and f 6 : for a b 6 f : for : for a b 8 : for a, b 6 6 f d C C d C 6 8 C C 8C 8 or C 5 : for an antiderivative of f : for including C : for antiderivative of f : for evaluating integral and equating it to 8 : for finding value of C f Copyright National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

7 . 985 AB/BC (a) Avea d Integration Techniques for the AB eam : for correct integral epression : for antidifferentiation and evaluation (b) Ave k sin b k k k k cos k k k k 98 k 9 k 7 d k 6 : for correct integral epression : for antidifferentiation : for setting Avea Aveb : for solving for k Copyright National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at