Integration Techniques for the AB exam
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1 For the AB 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
2 Complete this worksheet. 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 du u Trigonometric Functions: sin( udu ) cos( u) du sec ( udu ) csc ( udu ) sec( u) tan udu csc( u)cot udu Helpful to know: sin u tan( udu ) du cosu cosu cot( udu ) du sin u
3 Multiple Choice. (calculator not allowed) d 0 (A) ln (calculator not allowed) d (A) ln 8 ln ln 8 ln ln 8 ln ln. (calculator not allowed) 4 d (A) ln ln ln
4 4. (calculator not allowed) e d = (A) e e e e e e e e 5. (calculator not allowed) ( ) d = 0 (A) (calculator not allowed) d e (A) ln e C e C e C ln e C e C
5 7. (calculator not allowed) sin( ) cos( ) d (A) cos( ) sin( ) C cos( ) sin( ) C cos( ) sin( ) C cos( ) sin( ) C cos( ) sin( ) C 8. (calculator not allowed) cos d (A) sin C sin C sin C sin C 4 sin C 4 9. (calculator not allowed) cos d sin 4 (A) ln ln 4 ln ln ln e
6 0. (calculator not allowed) dy If sin cos d and if y = 0 when, what is the value of y when = 0? (A) 0. (calculator not allowed) If the substitution (A) u du u 4 4 u du u u du u u du 4u u du u u is made, the integral 4 d. (calculator not allowed) k If ( k ) d 8, then k 0 (A) 9 9 8
7 . (calculator not allowed) If f( c) d5 where c is a constant, then c f ( ) d c (A) 5 c 5 5 c c (calculator not allowed) b If f ( ) da b b, then ( ( ) 5) a f d a (A) a b 5 5b 5a 7b 4a 7b 5a 7b 6a 5. (calculator not allowed) If f is a linear function and 0 a b, then f ( ) d a (A) 0 ab b a b a b
8 6. (calculator not allowed) What are all values of k for which (A) 0 and, 0, and k d 0? 7, (calculator not allowed) (008 AB7) A particle moves along the -ais with velocity given by vt t 6tfor time t 0. If the particle is at position at time t 0, what is the position of the particle at time t? (A) (calculator allowed) If f is a continuous function and if F( ) f( ) for all real numbers, then f ( ) d (A) F() F() () F () F F(6) F() F(6) F() F(6) F() 9. (calculator allowed) On the graph of y f( ), the slope at any point (, y ) is twice the value of. If f (), what is the value of f ()? (A)
9 Free Response 0. (calculator not allowed) The figure above shows the graph of f, the derivative of a function f. The domain of f is the set of all such that 0. (a) Write an epression for f ( ) in terms of. (b) Given that f () 0, write an epression for f ( ) in terms of. (c) Sketch the graph of y f( ).
10 . (calculator not allowed) Let f be a function such that f ( ) 6 8. (a) Find f () if the graph of f is tangent to the line y at the point (0, ). (b) Find the average value of f ( ) on the closed interval [,].. (calculator not allowed) Let f be a differentiable function, defined for all real numbers, with the following properties. (i) f ( ) a b (ii) f () 6 and f () 8. (iii) f( ) d 8 (a) Find f ( ). Show your work.
11 . (calculator not allowed) Let f ( ) 4 and g( ) k sin for k 0. k (a) Find the average value of f on [, 4]. (b) For what value of k will the average value of g on [0, ] k be equal to the average value of f on [, 4].
Integration Techniques for the AB exam
For the AB 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
Integration Techniques for the BC exam
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
Integration Techniques for the BC exam
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
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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