CK12 Calculus - Single Variable
Alternatively, you may see the followingnotationsfor the aboveinverses,respectively, Sinceall trigonometricfunctionsare periodic ...
Hereare two otheridentitiesthat you may needto enterinto your calculator: The DerivativeFormulasof the InverseTrigonometricFunct ...
Example4: Differentiate Solution: Let so Example5: Find if Solution: Let The IntegrationFormulasof the InverseTrigonometricFunct ...
Example6: Evaluate Solution: Beforewe integrate,we use -substitution.Let (the squareroot of ). Then Substituting, Example7: Eval ...
Substituting To changethe limits, Thusour integralbecomes ReviewQuestions Find of Find of Find of Find of Find of Evaluate ...
Evaluate Evaluate Evaluate Giventhe points and find a point in the interval on the -axis that maximizes angle. Answers ...
Learnhow to find the limit of indeterminateform by L’Hospital’s rule. If the two functions and are both equalto zero at then t ...
Example2: Find Solution: We can see that the limit is when is substituted. UsingL’Hospital’s rule, Example3: Use L’Hospital’s ru ...
Solution: We can use L’Hospital’s rule sincethe limit producesthe once is substituted.Hence A broaderapplicationof L’Hospital’s ...
a. Showthat b. Use L’Hospital’s rule to find 7.Cauchy’s MeanValue Theoremstatesthat if the functions and are continuouson the in ...
7. IntegrationTechniques........................................................................................................ ...
rulesis neededto solvean integrationproblem. LearningObjectives A studentwill be able to: Computeby handthe integralsof a wide ...
Would you still expandthe integrandand then integrateterm by term?That wouldbe impracticaland time- consuming.A betterway of doi ...
Simplifying, TrigonometricIntegrands We can applythe changeof variabletechniqueto trigonometricfunctionsas long as is a differen ...
Integrating, Example4: This examplerequiresus to use trigonometricidentitiesbeforewe substitute.Evaluate Solution: Since , the i ...
Nowmakea changeof variable Choose Then or Substituting, This integralshouldlook obviousto you. The integrandis the derivativeof ...
As you may notice,the variable is still hangingthere.To write it in termsof since solving for we get, Substitutingbackinto the i ...
Thus ReviewQuestions In the followingexercises,evaluatethe integrals. ...
12. 13. 14. 15. Answers ...
13. 14. 15. IntegrationBy Parts LearningObjectives A studentwill be able to: Computeby handthe integralsof a wide varietyof fun ...
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