MATH Tutoril Indefinite Integrls The indefinite integrl f() d is to look for fmily of functions F () + C, where C is n rbitrry constnt, with the sme derivtive f(). Tble of Indefinite Integrls cf() d c f() d [f() + g()] d f() d + g() d k d k + C n d n+ n+ d ln + C e d e + C d ln + C + C if n sin d cos + C cos d sin + C sec d tn + C csc d cot + C sec tn d sec + C csc cot d csc + C + d tn + C d sin + C ( Emple. Evlute ( ) + 5/ + 5/ ) d. d ln tn 5 / Emple. If f () +, f() 4 nd f (), find f(). + C ln tn + + C. Integrte f () to give f (): f () ( + ) d + 6 4 + C. Put, f () C, thus f () + 6 4. Then integrte f () to give f(): f() ( + 6 4 ) d + 4 + C. Put, f() C 4, thus f() 4 + + 4. Prepred by Leung Ho Ming Homepge: http://mlhm.people.ust.hk/
The Fundmentl Theorem of Clculus Suppose f is continuous on [, b], then the function defined by F () f(t) dt is continuous on [, b] nd differentible on (, b), nd F () f(); b f() d F (b) F () where F is ny ntiderivtive of f, tht is, F f. This theorem gives the linkge between differentition nd integrtion. Emple. Find F () if F () Rewrite F () c ln c sin( + t ) dt + ln sin( + t ) dt. sin( + t ) dt c sin( + t ) dt ln sin( + t ) dt for some constnt c, differentiting with chin rule gives F () sin( + ( ) ) sin[ + ( ln ) ] sin( + ) + sin[ + (ln ) ]. Emple 4. Find f ( π g() ) if f() cos dt, where g() [ + sin(t )] dt. + t The derivtive of f is f () d g() d + t dt + [g()] g () d + [g()] d cos [ + sin(t )] dt + [g()] [ + sin(cos )] ( sin ) sin [ + sin(cos )]. + [g()] c Note tht g( π ) [ + sin(t )] dt, thus f ( π ). Emple 5. Evlute lim + t + t dt. lim + t + t dt lim + lim +. + + Emple 6. Evlute π/ sin θ + sin θ tn θ sec θ π/ dθ sin θ + sin θ tn θ sec θ π/ dθ. sin θ( + tn θ) sec θ dθ π/ sin θ dθ [ cos θ] π/ cos π + cos.
Emple 7. Suppose rcing cr ws running t 5 m/s. Strting from some instnce (t t ), decelertion of t m/s is pplied until the cr is totlly stopped. Find the brking distnce of the cr. Let T be the brking time, then the ccelertion of the cr is (t) t for t T. The velocity function v(t) cn be found from The brking time stisfies v(t ), so The brking distnce is then v(t) v() v(t) 5 t t (u) du u du v(t) 5 t. 5 T T. The brking distnce is m. s() s() v(t) dt ( ) 5 t dt [ 5t t 6. ] Substitution Rule for Indefinite Integrls If u g() is differentible function whose rnge is n intervl I, nd f is continuous on I, then [f[g()]g () d f(u) du. Emple 8. Evlute e d. Let u, then du 4 d, or d 4 du, thus e d 4 eu du 4 eu + C 4 e + C.
Emple 9. Evlute 4 + + d. + By long division, 4 + + + + + +. Thus 4 + + + where we used the subsitition u + in ( d + + ) d + ( + + + ) d + + u d + tn + ln u + tn + C + ln( + ) + tn + C, + d. Substitution Rule for Definite Integrls If g is continuous on [, b] nd f is continuous on the rnge of u g(), then b [f[g()]g () d g(b) g() f(u) du. Emple. Evlute thus d 5 + 8 d 5 + 8. Let u 5 + 8, then du 5 d. u when nd u 8 when, 8 [ ] 8 8 u u 5 du 5 u du 5 5 ( 8 ). Integrls of Symmetric Functions Suppose f is continuous on [, ]. If f is even, then If f is odd, then f() d f() d. Emple. Find the vlue of Write the integrl s f() d. (5 + ) 9 d. (5 + ) 9 d 5 9 d + 9 d. It is trivil to check tht 9 is n odd function. Thus 9 d. Netly, the grph of y 9 for is semicircle of rdius. Hence (5 + ) 9 d 5 9 d 5 π() 45π. 4
Emple. Let f be function stisfying f() 5 nd f () 6 + + sin. () Let g() f() 5. Show tht g( ) g() for ll. (b) Hence or otherwise evlute f() d. () g () f () 6 + sin. Using the fundmentl theorem of clculus, g() g() g (t) dt + t sin t dt + ( u) sin ( u) ( )du + u sin u du g (u) du [g( ) g()]. (u t) But g() f() 5, thus g() g( ), or g( ) g(). (b) From g() f() 5, we get f() g() + + 5. From (), g() is n odd function, thus Wheres + 5 is n even function, thus g() d. f() d ( + 5) d [g() + + 5] d g() d + ( + 5) d. Hence ( + 5) d [ + 5] 6. ( + 5) d 5
Eercise.. If f() stisfies ( + )f () + f() e + 4 for ll nd f() 5, find f(). [Hint: d d [( + )f()] ( + )f () + f().]. () Let f() be continuous on [, b], show tht there eists c (, b) such tht f(c) b [Hint: consider F () f(t) dt.] b f() d. (b) Let p() + + + n n be polynomil such tht + + + n. Prove tht there n + is number c (, ) such tht p(c). ( sin t. Find d ) + u4 d du dt. 4. Let F () (4 t ) dt for. Find (F ) (7). 5. Suppose >, for wht vlue of would 6. The figure shows the grph of y f(), where. y ln d ttins the minimum vlue? 9 4 5 6 7 8 9 Find where do the bsolute mimum nd bsolute minimum of F () mimum nd minimum vlues. f(t) dt occur nd the corresponding 6