![]() ![]() ∫tan 35 x sec x dx ( ) ( ) ( )Ī ∫ f x dx represents the net area between fx ( ) and the Each integral will beĭealt with differently. To tangents using sec 22 xx = +1 tan, thenĤ. Tan 22 xx =sec − 1, then use the substitutionĢ. For∫tan nmx sec x dx we have the following :ġ. Integral into a form that can be integrated. To sines using cos 22 xx = −1 sin, then use Strip 1 sine out and convert rest toĬosines using sin 22 xx = −1 cos, then useĢ. Products and (some) Quotients of Trig Functions For ∫sin nmx cos x dx we have the following :ġ. X −− xxdx =−+ = x −− xxdx −−+ x − xc ∫∫ eeeee Choose u and dv from integral and compute du by differentiating u and compute v using v =∫ dv. = − ∫∫ Integration by Parts : ∫∫ u dv uv = − v du andībb ∫∫ aau dv uv = a − v du. ![]() For indefinite integrals drop the limits of integration. u Substitution : The substitution u gx = ( )will convert ( )( ) ( ) ( ) Note that at many schools all but the Substitution Rule tend to be taught in a Calculus II class. Sec 2 u du =tan u c + ∫ ∫sec tan u u du =sec u c + ∫csc cot u udu =−+csc u cĬsc 2 u du =−+cot u c ∫ ∫tan u du =ln sec u + c ∫sec u du = ++ln sec u tan u c ∫ au 22 + 1 du = aa 1 tan− 1 ( ) u + c 221 sin 1 ( ) ua Uudu = + c ∫ ee ∫cos u du =sin u c + ∫sin u du =−+cos u c # If m fx M ≤≤( ) on a xb ≤≤ then ( ) ( ) ( )ġ1 ln ∫ ax b + dx = ++ a ax b c ∫ln u du = u ln( ) u −+ u c # If f x gx ( ) ( )≥ on a xb ≤≤then ( ) ( ) Properties ∫ ∫∫ f x ( ) ( )±= ± g x dx f x dx ( ) g x dx ( )ī bb ∫ ∫∫ af x ±= ± g x dx aaf x dx g x dxīa ∫∫ abf x dx = − f x dx ∫∫ cf x dx ( ) = c f x dx ( ), c is a constantīb ∫∫ aacf x dx = c f x dx, c is a constantī cb ∫∫∫ a acf x dx = f x dx + f x dx for any value of c. # Part II : fx ( )is continuous on ab, , Fx ( ) is an anti-derivative of fx ( )( i. ![]() # Part I : If fx ( ) is continuous on ab, thenĪ g x =∫ f t dt is also continuous on ab , # where Fx ( ) is an anti-derivative of fx ( ). Indefinite Integral : ∫ f x dx ( ) = F x ( )+ c # is a function, Fx ( ), such that F x fx ′( ) ( )=. # Anti-Derivative : An anti-derivative of fx ( ) Width ∆ x and choose xi * from each interval. Remember to leave feedback and you will earn points toward FREE TPT purchases.# Definite Integral: Suppose fx ( ) is continuous ![]() You might also be interested in the Just 4 Things FREEBIE which will help your students practice these techniques. Students will learn to use these valuable techniques for reducing their time on the AP Exam calculator prompt questions. This short instructional page will be a great addition to your AP Calculus students’ notebooks. integration using the numeric integration feature on the calculator.storing values as variables for use in calculations, and.finding bounds(points of intersection),.Graphing a function in an arbitrary window. Your AP Calculus students are expected to know how to do just 4 things on the AP Exam.ġ. This is a free TI-84 Plus CE™ Calculator Cheat Sheet. ![]()
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