Worksheet Week 7 Section
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1 Worksheet Week 7 Section This worksheet is for improvement of your mathematical writing skill. Writing using correct mathematical epression and steps is really important part of doing math. Please print out this worksheet and try to solve problems, following given steps. You don t need to submit this worksheet. It is not homework. The steps in the solution represent important information you have to use in the solution. It must be mentioned on your own solution.. (Preliminaries) By using trigonometric identities, we can show that when is a trigonometric function, then some formula involving is another trigonometric function. The followings are several eamples. (a) Let a sin t for some positive constant a > 0 and t. Show that a a cos t. a a (a sin t) a a sin t a ( sin t) a cos t a a cos t a cos t a cos t a cos t ( t cos t 0) (b) Let a tan t for some positive constant a > 0 and < t <. Show that a + a sec t. a + a + (a tan t) a + a tan t a ( + tan t) a sec t a + a sec t a sec t a sec t a sec t ( < t < sec t cos t > 0)
2 Compare to above identities, the net one is delicate. (c) Let a sec t for some positive constant a > 0 and 0 t < or < t. Show that a { a tan t, 0 t <, a tan t, < t. a (a sec t) a a sec t a a (sec t ) a tan t a a tan t a tan t a tan t { a tan t, 0 t < a tan t,, < t Note that if > a then sec t > 0 so 0 t <. On the other hand, if < a, then sec t < 0 and sign of. < t. Therefore we can pick the range of t from the By using these identities, we can compute an integral involving a, a +, a. s you can see, this is precisely the inverse of the substitution method.. Evaluate the integral 9 d. (a) Find a part of given function which we can epress both and that part as trigonometric functions with respect to t. Use above identities. sin t 9 cos t (b) Compute d in terms of dt. d dt cos t d cos tdt (c) Substitute given integral as an integral with respect to t. ( sin t) d 9 cos t cos tdt 7 sin tdt
3 (d) Evaluate given integral. 7 sin tdt 7 sin t sin tdt 7( cos t) sin tdt 7( u )( )du (substitute by u cos t) 7(u )du 7( u u) + C 9u 7u + C 9 cos t 7 cos t + C (e) Replace t by using. cos t 9 ( ) 9 9 cos 9 t 7 cos t + C C ( 9 ) C In this step, sometimes you need inverse trigonometric functions.. Evaluate the integral d d. d 9( 9 + ) d 9 + d 9 + tan t, 9 + sec t d dt sec t d sec tdt d 9 + sec t sec tdt sec tdt ln sec t + tan t + C ln C
4 When we have a rational function, the computation of its antiderivative heavily depends on finding partial fractions, a sum of simpler fractions which is same with original fraction. Here is a general method to find partial fractions. 4. Epand the fraction by partial fractions (a) Factor out the denominator ( + )( + ) (b) In this problem, there is no repeated factor. Then the partial fraction is of the form + + B +. By taking common denominator, find equations for and B. + + B ( + ) B( + ) ( + ) + B( + ) + + ( + )( + ) ( + )( + ) ( + )( + ) ( + B) + + B ( + )( + ) ( + )( + ) ( + B) + + B + B, + B 0 (c) Solve the equation and find and B. ( + B) ( + B) 0 ( B ), B (d) Find the partial fraction ( + ) + ( + ) 4
5 5. Epand the fraction by partial fractions (a) Factor out the denominator. + + ( + ) (b) In this problem, there is a repeated factor +. Then the partial fraction is of the form + + B ( + ). By taking common denominator, find equations for and B. + + B ( + ) ( + ) ( + ) + B ( + ) + B ( + ) ( + ) + + B ( + ) + ( + ), + B (c) Solve the equation and find and B., + B B (d) Find the partial fractions for given function ( + ) 5
6 6. Epand the fraction by partial fractions. + + ( )( ) + B ( ) B( ) ( ) + B( ) + ( )( ) ( )( ) ( )( ) ( + B) B ( )( ) ( )( ) + B, B 0 ( + B) + ( B) + 0 B ( ) + + 6
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