Agenda Feb. 8HW Questions• Lesson 7.2 – More Trig Integrals

• Partner Quiz over 7.1-7.2 MONDAY• Due today: Challenge Problem 1

"Science is what we understand well enough to explain to a computer. Art is everything else we do." -- Donald Knuth

First, a review of an important trig property you’ll be using here:

Pythagorean Identity

Recall:

∫ 𝑠𝑒𝑐2( x)𝑑𝑥 ∫ sec 𝑥 ∙𝑡𝑎𝑛 𝑥 𝑑𝑥

∫ tan(𝑥 )𝑑𝑥 ∫ sec (𝑥 )𝑑𝑥

Goal: Get into one of these forms

∫𝑡𝑎𝑛𝑝 (𝑥 )𝑠𝑒𝑐 2(x)𝑑𝑥 ∫ 𝑠𝑒𝑐𝑝 𝑥 (sec 𝑥 tan𝑥 )𝑑𝑥

∫ 𝑠𝑒𝑐𝑚 𝑥𝑡𝑎𝑛𝑛 𝑥 𝑑𝑥

orWhich one? Depends on whether secant is even or odd!

If power of secant is even

∫𝑡𝑎𝑛3 (𝑥 )𝑠𝑒𝑐4(x )𝑑𝑥

∫ 𝑠𝑒𝑐𝑚 𝑥𝑡𝑎𝑛𝑛 𝑥 𝑑𝑥Save out a . Convert the remaining secants using:

If power of secant is odd

∫𝑡𝑎𝑛3 (𝑥 )𝑠𝑒𝑐3 (x)𝑑𝑥

∫ 𝑠𝑒𝑐𝑚 𝑥𝑡𝑎𝑛𝑛 𝑥 𝑑𝑥Save out a . Convert the remaining tangents using:

Only has tangent

∫𝑡𝑎𝑛4 (𝑥 )𝑑𝑥

∫𝑡𝑎𝑛𝑛𝑥 𝑑𝑥Replace a with May require multiple applications of this property

Only has secantNo tangent

∫ 𝑠𝑒𝑐 4 (𝑥 )𝑑𝑥

∫ 𝑠𝑒𝑐𝑚 𝑥 𝑑𝑥If even: Same as earlier…Save out a . Convert the remaining secants using:

Only has secantNo tangent ∫ 𝑠𝑒𝑐𝑚 𝑥 𝑑𝑥

If odd: use integration by parts.

∫ 𝑠𝑒𝑐3 (𝑥 )𝑑𝑥