Integration Techniques for the BC exam

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1 Integration Techniques for the B eam For the B eam, students need to: determine antiderivatives of the basic functions calculate antiderivatives of functions using u-substitution use algebraic manipulation to rewrite the integrand prior to integrating use trigonometric identities and properties of eponents and logarithms to rewrite solutions use geometric interpretations of the definite integral integration by parts integration by partial fractions (non-repeating linear factors only) improper integrals (as limits of definite integrals) opyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

2 Integration Techniques for the AB eam omplete this worksheet as a review of the antiderivatives of the basic functions. These rules should be memorized. Basic Integration kf( udu ) [ f ( u) g( u)] du du n u du du u u a du u e du Inverse Trigonometric du u Trigonometric Functions: sin( udu ) cos( udu ) sec ( udu ) csc ( udu ) sec( u) tan udu csc( u)cot udu Helpful to know: tan( udu ) cot( udu ) du u opyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

3 Integration Techniques for the B Eam Multiple hoice: (Questions 9 are appropriate for AB). (calculator not allowed) 4 d ln () ln (D) ln. (calculator not allowed) d e ln e e () e (D) ln e e. (calculator not allowed) cos d sin sin sin sin 4 sin 4 () (D) opyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

4 4. (calculator not allowed) cos d sin 4 Integration Techniques for the B Eam ln ln 4 () ln (D) ln ln e 5. (calculator not allowed) dy If sin cos d and if y = 0 when, what is the value of y when = 0? () 0 (D) 6. (calculator not allowed) If the substitution () (D) u du u 4 4 u du u u du u u du 4u u du u u is made, the integral 4 d opyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

5 7. (calculator not allowed) If f( c) d5 where c is a constant, then Integration Techniques for the B Eam c f ( ) d c 5 c 5 () 5 c (D) c (calculator not allowed) b If f ( ) da b b, then ( ( ) 5) a f d a ab 5 5b 5a () 7b 4a (D) 7b 5a 7b 6a 9. (calculator allowed) If f is a continuous function and if F( ) f( ) for all real numbers, then f ( ) d F() F() () F () F () F(6) F() (D) F(6) F() F(6) F() opyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

6 0. (calculator not allowed) e d Integration Techniques for the B Eam () (D) e e 4 e e e e 4 e e e 4. (calculator not allowed) sec d tan tan () sec sec tan (D) tan ln cos tan ln cos. (calculator not allowed) If cos d f( ) sin d, then ( ) sin cos sin () cos sin (D) 4cos sin cos4sin f opyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

7 . (calculator not allowed) If f ( ) sin( ) d f( )cos cos d, then ( ) () (D) sin cos Integration Techniques for the B Eam f could be 4. (calculator not allowed) 0 f ( ) 4 f ( ) 6 g ( ) 4 g ( ) The table above gives values of f, f, g, and g for selected values of. If f( ) g( ) d 5, then f ( g ) ( d ) 0 4 () (D) opyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

8 5. (calculator not allowed) d d ( )( ) Integration Techniques for the B Eam ln ln () ln ( )( ) ln ln (D) ln ( )( ) 6. (calculator not allowed) 7 d ()( ) ln ln ln ln () ln ln (D) 6 () ( ) () ( ) opyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

9 7. (calculator not allowed) ( ). d is 0 Integration Techniques for the B Eam ln ln () ln (D) ln divergent 8. (calculator not allowed) d is ln () (D) noneistent 9. (calculator not allowed) d is () 9 7 (D) 9 7 noneistent opyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

10 0. (calculator allowed) Let F() be an antiderivative of ln Integration Techniques for the B Eam. If F() = 0, then F(9) = () 5.87 (D).08, (calculator allowed) If f is the antiderivative of () 0.06 (D) such that f () 0, then f (4) opyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

11 Free Response Integration Techniques for the B Eam. (calculator not allowed) 5 (a) Determine e d. (b) Using integration by parts, derive a general formula for n k e d, k 0, in which the n resulting integral involves. opyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

12 Integration Techniques for the B Eam. (calculator not allowed) Let g be the function given by g ( ). (d) The average value of a function f on the unbounded interval [ a, ) is defined to be b f ( d ) a lim. Show that the improper integral b b a gd ( ) is divergent, but the 4 average value of g on the interval [4, ) is finite. opyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

13 4. (calculator not allowed) Determine whether or not reasoning. Integration Techniques for the B Eam e d converges. If it converges, give its value. Show your 0 opyright 0 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at

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