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int(sec(x)+tan(x))dx

find integration by subtutitions.

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George C. | Humboldt State and Georgetown graduateHumboldt State and Georgetown graduate
5.0 5.0 (2 lesson ratings) (2)
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= (1/cos x   +    sin x/cos x) dx

= (1 + sin x)/cos x   dx

= (1 + sin x)(1 - sin x)/(cos x)(1 - sin x)   dx

= (1 - (sin x)^2)/(cos x)(1 - sin x)  dx

= (cos x)^2/(cos x)(1 - sin x)  dx 

= cos x/(1 - sin x)  dx

Integrate:

- ln (1 - sin x)

 

Robert J. | Certified High School AP Calculus and Physics TeacherCertified High School AP Calculus and Ph...
4.6 4.6 (13 lesson ratings) (13)
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∫(sec(x)+tan(x))dx

= ∫ 1/(sec(x) - tan(x)) dx, after multiplying the top and the bottom by sec(x) - tan(x) and using the identity sec^2(x) = tan^2(x) + 1

= ∫cos(x)/(1 - sin(x)) dx

= ∫-1/(1 - sin(x)) d(1- sin(x)), mental substitution, or you can let u = 1-sin(x)

= -ln(1-sin(x)) + c