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For more information, please see full course syllabus of Calculus AB
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Lecture Comments (2)

1 answer

Last reply by: Kishan Patel
Wed Feb 15, 2017 11:01 PM

Post by Derui Wang on February 15 at 07:28:20 AM

how to solve for integrals like tanaxsecbx?

Trigonometric Rules of Integrals

  • Memorize!
  • Careful not to confuse with trig derivative rules
  • Useful especially during multiple choice portion of AP exam

Trigonometric Rules of Integrals

∫sin5x dx
  • ∫sin5x dx = −cos5x ([1/5]) + c
∫sin5x dx = −[1/5] cos5x + c
∫cos[x/3] dx
  • ∫cos[x/3] dx = sin[x/3] [1/([1/3])] + c
∫cos[x/3] dx = 3 sin[x/3] + c
∫sec2 x dx
∫sec2 x dx = tanx + c
∫sec3x tan3x dx
  • ∫sec3x tan3x dx = sec3x [1/3] + c
∫sec3x tan3x dx = [1/3] sec3x + c
∫sec2 x + sec3x tan3x dx
  • ∫sec2 x + sec3x tan3x dx = ∫sec2 x dx + ∫sec3x tan3x dx
  • ∫sec2 x + sec3x tan3x dx = tanx + sec3x [1/3] + c
∫sec2 x + sec3x tan3x dx = tanx + [1/3] sec3x + c
∫csc2 [x/2] dx
  • This is close to the derivative of cot[x/2], but not quite.
  • ∫csc2 [x/2] dx = (−1)(−1)∫csc2 [x/2] dx
  • ∫csc2 [x/2] dx = − ∫− csc2 [x/2] dx
  • ∫csc2 [x/2] dx = −(cot[x/2]) [1/([1/2])] + c
∫csc2 [x/2] dx = −2cot[x/2] + c
−∫cscx cotx dx
  • −∫cscx cotx dx = ∫− cscx cotx dx
−∫cscx cotx dx = cscx + c
∫3 sin[x/3] dx
  • ∫3 sin[x/3] dx = 3 ∫sin[x/3] dx
  • ∫3 sin[x/3] dx = 3 (−cos[x/3] [1/([1/3])]) + c
∫3 sin[x/3] dx = − 9 cos[x/3] + c
∫sin2x − cos3x dx
  • ∫sin2x − cos3x dx = ∫sin2x dx − ∫cos3x dx
∫sin2x − cos3x dx = [1/2] sin2x − [1/3] cos3x + c
∫secx tanx − cscx cotx dx
  • ∫secx tanx − cscx cotx dx = ∫secx tanx dx − ∫cscx cotx dx
  • ∫secx tanx − cscx cotx dx = ∫secx tanx dx + ∫− cscx cotx dx
∫secx tanx − cscx cotx dx = secx + cscx + c

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Trigonometric Rules of Integrals

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Trigonometric Rules 0:09
    • Integral of SIN
  • Example 1: Integral of SIN 1:46
  • Example 2: Integral of COS 2:38
  • Example 3: With 2 terms of X 3:06
  • Example 4: Integral of SEC 4:15
  • Example 5: Integral of CSC 5:06
  • Example 6 6:18