{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

7.3 - 7.3 Trigonometric Substitution(Dated To make the...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
7.3: Trigonometric Substitution (Dated: October 20, 2011) To make the trigonometric substitutions x = a sin θ , x = a tan θ , and x = a sec θ , one needs to restrict θ such that x = x ( θ ) is one-to-one. Also, these trigonometric substitutions are effective for different radical expressions in view of the specified trigonometric identities. 1. x = a sin θ is effective for the radical expression p a 2 - x 2 because 1 - sin 2 θ = cos 2 θ . One usually re- stricts θ to the intervals - π 2 < θ π 2 or - π 2 θ < π 2 . 2. x = a tan θ is effective for the radical expression p a 2 + x 2 because 1 + tan 2 θ = sec 2 θ . One usually re- stricts θ to the interval - π 2 < θ < π 2 . 3. x = a sec θ is effective for the radical expression p x 2 - a 2 because sec 2 θ - 1 = tan 2 θ . One usually re- stricts θ to the intervals 0 θ < π 2 or π θ < 3 π 2 . Example 1 [Exercise 7.3.6 in the text] Let x = sec θ , 0 θ < π 2 . Then p x 2 - 1 x = p sec 2 θ - 1 sec θ = p tan 2 θ sec θ = | tan θ | sec θ = tan θ sec θ and dx = sec θ tan θ d θ . It follows that Z p x 2 - 1 x dx = Z | tan θ | sec θ sec θ tan θ d θ = Z tan 2 θ d θ = Z (sec 2 θ - 1) d θ = tan θ - θ + C .
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern