# If a1 you have 1 c u sin u 1 du1 2 2 cu tan u 1 du1 2

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Chapter 1 / Exercise 16
Calculus: Early Transcendental Functions
Edwards/Larson
Expert Verified
If a=1, you have: 1. Cusinu1du122. Cutanu1du123. Cuuudu12sec1
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Chapter 1 / Exercise 16
Calculus: Early Transcendental Functions
Edwards/Larson
Expert Verified
Identifying Patterns For each of the integrals below, which inverse trig function is involved? 82 241316dxx2254dxxx29dxx
Warning Many integrals look like the inverse trig forms Which of the following are of the inverse trig forms? 83 21dxx21x dxx21dxx21x dxxIf they are not, how are they integrated?
Try These Look for the pattern or how the expression can be manipulated into one of the patterns 84 28116dxx2125x dxx24415dxxx251016xdxxx
Completing the Square Often a good strategy when quadratic functions are involved in the integration Remember … we seek (x –b)2+ c Which might give us an integral resulting in the arctan function 2210dxxx
Completing the Square Try these 222413dxxx224dxxx
Example : Evaluate the following integrals using the formulas for integrals yielding inverse trigonometric functions: 252.1sds22.2xxdx23.32xxdx916.42yydy1.52xxxdx3423.62xxxdx32213.72xxdxx
HOMEWORK:Evaluate the following integral. 88 4916.1rrdrxxdx2cos2sin.2xxdx1.3dxxx2241.4dxeexx21.5223.6xxxdxd2sin1cos.73422.823xxdxx2ln1.9xxdx19.102xxdx1022.11tdt12/12443.12dxxx3ln2ln.13zzeedz
Integration of Hyperbolic FunctionsTOPIC
OBJECTIVES identify the different hyperbolic functions; find the integral of given hyperbolic functions; determine the difference between the integrals of hyperbolic functions; and evaluate integrals involving hyperbolic functions.
2sinh.1xxeex2cosh.2xxeexxxxxeeeexxxcoshsinhtanh.3xxxxeeeexxtanh1coth.4xxeexhx2cosh1sec.5xxeexhx2sinh1csc.6Definitions:
Differentiation Formulasuduudcoshsinh.1uduudsinhcosh.2uduhud2sectanh.3uduhud2csccoth.4uduhuhudtanhsecsec.5uducothhucschucscd.6
. Note: The hyperbolic functions are defined in terms of the exponential functions. Its differentials may also be found by differentiating its equivalent exponential form. Similarly, the integrals of the hyperbolic functions can be derived by integrating the exponential form equivalent.
Integration FormulasCuuducoshsinh.1Cuudusinhcosh.2Cuuduhtanhsec.32Cuduhcothcsc.42Chuuduhusectanhsec.5Chucscuducothhucsc.6uuduuducoshsinhtanh.7cucoshlnuuduudusinhcoshcoth.8cusinhlnCehuduu1tan2sec.9Cusinhtan1or11lncsc.10CeehuduuuC1ucosh1ucoshln217.
Hyperbolic Functions Trigonometric Functions 1xsinhxcosh22xhsecxtanh122xhcsc1xcoth22ysinhxcosh