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 definitions 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 04 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org
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( u ) [ f ( u) g( u)] n u u u a u e Inverse Trigonometric u u Trigonometric Functions: sin( u ) cos( u ) sec ( u ) csc ( u ) sec( u) tan u csc( u)cot u Helpful to know: sin u tan( u ) cosu cosu cot( u ) sin u opyright 04 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org
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 04 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org
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 u 4 4 u u u u u 4u u u u is made, the integral 4 d opyright 04 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org
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 5 5 8. (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 04 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org
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 04 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org
. (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) 7 5 0 opyright 04 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org
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 04 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org
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 4 9 7 7 () 9 7 (D) 9 7 noneistent opyright 04 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org
Integration Techniques for the B Eam 0. (calculator allowed) If f is the antiderivative of 0.0 0 () 0.06 (D) 0.76 0.69 5 such that f () 0, then f (4) opyright 04 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org
Free Response 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 04 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org
. (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. (calculator not allowed) The derivative of a function f is given by f ( ) ( ) e for 0 and f () 7. (c) Find the value of f ()., opyright 04 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org