#DESCRIPTION # Trigonometric Substitution. ## sbrummel tagged and PAID on 2-20-2004 ## DBchapter('Techniques of Integration') ## DBsection('Trig Substitution') ## Date('6/3/2002') ## Author('Arnie Pizer') ## Institution('rochester') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('7.2') ## Problem1('1,2,3') #ENDDESCRIPTION #KEYWORDS('Substitution' , 'Trig Substitution' , 'Trigonometric Substitution' ) DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); # No partial credit on this problem, so we say: install_problem_grader(~~&std_problem_grader); TEXT(&beginproblem); $showPartialCorrectAnswers = 0;$a = random(2,9,1); $aa =$a**2; qa(~~@questions, ~~@answers, "$$\displaystyle \int x^2 \sqrt{aa + x^2} \, dx$$" , "A", "$$\displaystyle \int \frac{x^2 \, dx}{\sqrt{aa - x^2}}$$" , "B", "$$\displaystyle \int \sqrt{x^2 - aa} \, dx$$" , "C", "$$\displaystyle \int \frac{dx}{(aa + x^2)^3}$$" , "A", "$$\displaystyle \int \frac{dx}{(aa - x^2)^{3/2}}$$" , "B", "$$\displaystyle \int (x^2 - aa)^{5/2} \, dx$$" , "C" ); # Now randomize the questions: @slice = &NchooseK(6,5); @shuffle = &shuffle(scalar(@slice)); ($ans1,$ans2, $ans3,$ans4, $ans5) = @answers[@slice] ; TEXT(EV3(<<'EOT')); For each of the indefinite integrals below, choose which of the following substitutions would be most helpful in evaluating the integral. Enter the appropriate letter (A,B, or C) in each blank. DO NOT EVALUATE THE INTEGRALS.$BR $BR For obvious reasons, you are limited to 4 attempts on this problem.$BR $BR A. $$x = a \tan \theta$$$BR B. $$x = a \sin \theta$$ $BR C. $$x = a \sec \theta$$$BR $BR EOT TEXT(EV2(match_questions_list_varbox(5,@questions[@slice]))); &ANS( std_str_cmp($ans1), std_str_cmp($ans2), std_str_cmp($ans3), std_str_cmp($ans4), std_str_cmp($ans5)); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ ##DESCRIPTION ##KEYWORDS('integrals', 'trig', 'substitution') ## sbrummel tagged and PAID on 2-20-2004 ## DBsubject('Calculus') ## DBchapter('Techniques of Integration') ## DBsection('Trig Substitution') ## Date('6/3/2002') ## Author('Arnie Pizer') ## Institution('rochester') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('7.2') ## Problem1('26,28') ##ENDDESCRIPTION DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem()); $showPartialCorrectAnswers = 1;$a = random(2,9,1); $k = random(4,9,1);$aa = $a*$a; $n = 2*$k + 3; $m = 2*$k + 1; $p = 2*$k; $ans = "(1/($aa*$m)) * (x^2/($aa - x^2))**($m/2)"; TEXT(EV2(<"antider")); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ ##DESCRIPTION ##KEYWORDS('integrals', 'trig', 'substitution') ## sbrummel tagged and PAID on 2-20-2004 ## DBsubject('Calculus') ## DBchapter('Techniques of Integration') ## DBsection('Trig Substitution') ## Date('6/3/2002') ## Author('Arnie Pizer') ## Institution('rochester') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('7.2') ## Problem1('27') ##ENDDESCRIPTION DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem());$showPartialCorrectAnswers = 1; $a = random(2,9,1);$aa = $a*$a; $ans = "(1/(2*$a**3)) * (arctan(x/$a) + ($a * x)/($aa + x^2))"; TEXT(EV2(<"antider")); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ ##DESCRIPTION ##KEYWORDS('integrals', 'trigonometric','substitution') ## sbrummel tagged and PAID on 2-20-2004 ## DBsubject('Calculus') ## DBchapter('Techniques of Integration') ## DBsection('Trig Substitution') ## Date('6/3/2002') ## Author('Arnie Pizer') ## Institution('rochester') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('7.2') ## Problem1('1') ##ENDDESCRIPTION DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem());$showPartialCorrectAnswers = 1; $a = random(1,20,1);$a2 = $a*$a; $funct = "($a2-x**2)**.5/(-$a2 * x)"; TEXT(EV2(<"antider")); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ ##DESCRIPTION ##KEYWORDS('integrals', 'trigonometric','substitution') ## sbrummel tagged and PAID on 2-20-2004 ## DBsubject('Calculus') ## DBchapter('Techniques of Integration') ## DBsection('Trig Substitution') ## Date('6/3/2002') ## Author('Arnie Pizer') ## Institution('rochester') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('7.2') ## Problem1('13') ##ENDDESCRIPTION DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem());$showPartialCorrectAnswers = 1; $a = random(1,20,1);$b = random(1,20,1); $b2 =$b*$b;$a2 = $a*$a; $lend =$b/$a;$rend = $lend +1;$funct = "($a2* x**2-$b2)**.5 -$b*arccos($lend/x)"; TEXT(EV2(<[$lend,$rend], mode=>"antider", vars=>"x")); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ ##KEYWORDS('integrals', 'trigonometric', 'substitution') ##DESCRIPTION ## Evaluate a definite integral ##ENDDESCRIPTION ## AmberHolden tagged ## Shotwell cleaned ## DBsubject('Calculus') ## DBchapter('Techniques of Integration') ## DBsection('Trig Substitution') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('7.3') ## Problem1('12,22') DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PGbasicmacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem()); $showPartialCorrectAnswers = 1;$a = random(2,3,1); $b = random(2,5,1);$c = $a *$a; $s = arctan(1);$soln = "1/$a *$b * $s"; BEGIN_TEXT Evaluate the definite integral.$BR $\int_{0}^{\frac{1}{a}} \frac{b}{1 + c x^2} dx$ $BR Answer: \{ans_rule( 30) \}$BR END_TEXT ANS(num_cmp($soln)); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ ##DESCRIPTION ##KEYWORDS('integrals', 'inverse trig functions') ## sbrummel tagged and PAID on 2-20-2004 ## DBsubject('Calculus') ## DBchapter('Techniques of Integration') ## DBsection('Trig Substitution') ## Date('6/3/2002') ## Author('Arnie Pizer') ## Institution('rochester') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('7.2') ## Problem1('20,23') ##Ellis & Gulick section 6.5 ##Authored by Zig Fiedorowicz 1/29/2000 ##ENDDESCRIPTION DOCUMENT(); loadMacros( "PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" );$showPartialCorrectAnswers = 1; $aa = random(3,8,1);$bb = random(5,10,1); if ($aa ==$bb) {$aa++;}$a2 = $aa**2;$b2 = $bb**2; TEXT(beginproblem()); BEGIN_TEXT $$\displaystyle \int \frac{1}{\sqrt{a2 - b2 x^2}}dx$$ = \{ ans_rule(45)\} +$$C$$$PAR WeBWorK notation for $$\sin^{-1}(x)$$ is arcsin(x) or asin(x), for $$\tan^{-1}(\ x)$$ it's arctan(x) or atan(x). END_TEXT ##set $PG_environment{'textbook'} in webworkCourse.ph if (defined($textbook)) { if ($textbook eq "EllisGulick5") { BEGIN_TEXT$PAR This is similar to problems 39-42 of Section 6.5 of the text. END_TEXT } } ANS(fun_cmp("arcsin($bb*x/$aa)/$bb", limits=>[0,$aa/$bb-.00001], mode=>"antider", vars=>"x")); ENDDOCUMENT(); ________________________________________________________________________________ ##DESCRIPTION ##KEYWORDS('integrals', 'inverse trig functions') ## sbrummel tagged and PAID on 2-20-2004 ## DBsubject('Calculus') ## DBchapter('Techniques of Integration') ## DBsection('Trig Substitution') ## Date('6/3/2002') ## Author('Arnie Pizer') ## Institution('rochester') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('7.2') ## Problem1('27') ##Ellis & Gulick section 6.5 ##Authored by Zig Fiedorowicz 1/29/2000 ##ENDDESCRIPTION DOCUMENT(); loadMacros( "PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" );$showPartialCorrectAnswers = 1; $a = random(2,6,1);$b = random(5,10,1); $a2 = 2*$a; $c =$a**2 + $b**2; TEXT(beginproblem()); BEGIN_TEXT $$\displaystyle \int \frac{1}{x^2+a2 x + c}\,dx$$ = \{ ans_rule(45)\} +$$C$$$PAR WeBWorK notation for $$\sin^{-1}(x)$$ is arcsin(x) or asin(x), for $$\tan^{-1}(\ x)$$ it's arctan(x) or atan(x). END_TEXT ##set $PG_environment{'textbook'} in webworkCourse.ph if (defined($textbook)) { if ($textbook eq "EllisGulick5") { BEGIN_TEXT$PAR This is similar to problems 36-38 of Section 6.5 of the text. END_TEXT } } ANS(fun_cmp("(1/$b)*arctan((x+$a)/$b)", mode=>"antider", vars=>"x")); ENDDOCUMENT(); ________________________________________________________________________________ ##DESCRIPTION ##KEYWORDS('integrals', 'trigonometric','substitution') ## sbrummel tagged and PAID on 2-20-2004 ## DBsubject('Calculus') ## DBchapter('Techniques of Integration') ## DBsection('Trig Substitution') ## Date('6/3/2002') ## Author('Arnie Pizer') ## Institution('rochester') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('7.2') ## Problem1('23') ##ENDDESCRIPTION DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PG.pl", "PGbasicmacros.pl", "PGchoicemacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); TEXT(beginproblem());$showPartialCorrectAnswers = 1; $a = random(1,15,1);$a1 = $a+1;$a2 = 2*$a;$funct = "(x-$a)*(($a2*x-x**2)**.5)/2 + $a*$a*arcsin((x-$a)/$a)/2"; TEXT(EV2(<[$a,$a1], mode=>"antider", vars=>"x")); ENDDOCUMENT(); # This should be the last executable line in the problem. ________________________________________________________________________________ ##KEYWORDS('integrals', 'trigonometric' 'substitution') ##DESCRIPTION ## Evaluate an indefinite integral ##ENDDESCRIPTION ## AmberHolden tagged ## Shotwell cleaned ## DBsubject('Calculus') ## DBchapter('Techniques of Integration') ## DBsection('Trig Substitution') ## Date('6/3/2002') ## Author('') ## Institution('') ## TitleText1('Calculus Early Transcendentals') ## EditionText1('4') ## AuthorText1('Stewart') ## Section1('7.3') ## Problem1('27') DOCUMENT(); loadMacros( "PGbasicmacros.pl", "PGanswermacros.pl", "PGauxiliaryFunctions.pl" ); $showPartialCorrectAnswers = 1;$a = random(2,6,1); $b = random(5,10,1);$a2 = 2*$a;$c = $a**2 +$b**2; TEXT(beginproblem()); BEGIN_TEXT $BR $$\displaystyle \int \frac{1}{x^2+a2 x + c}\,dx$$ = \{ ans_rule(45)\} $$+$$ $$C$$ END_TEXT ANS(fun_cmp("(1/$b)*arctan((x+$a)/$b)",mode=>'antider')); ENDDOCUMENT();