Integral Of Trig Functions

Online Tutoring Is The Easiest, Most Cost-Effective Way For Students To Get The Help They Need Whenever They Need It.

SIGN UP FOR A FREE TRIAL


Some integration is the inverse differentiation; the integrals of some elementary functions follow from the standard results of differential calculus. Following are the few examples of integral trigonometric functions.

Question: - Evaluate
  1. ∫ cos ^2 2 x dx
  2. ∫ sin ^3 x dx
  3. ∫ sin 3 x sin 4 x d x
Solution: -∫ cos ^2 2 x dx = ½  ∫ 2 cos ^2 2x dx
 
 = ½ ∫ (1 + cos 4 x)

 = ½ [∫ dx + ∫ cos 4x]

 = ½ [ x + (sin 4x)/4] + c                     Where c is a constant.
 
2.Since, sin 3 x = 3 sin x – 4 sin ^3 x,

  Therefore, 4 sin ^3 x = 3 sin x – sin 3 x

 Hence, ∫ sin ^3 x dx = ¼ ∫ 4 sin ^3 x dx

 = ¼ ∫ (3 sin x – sin 3 x) dx

 = ¼ [3 ∫ sin x dx - ∫ sin 3 x dx]

 = ¼ [3 (- cos x) – (- cos 3 x / 3) ] + k

 = -3/4 cos x + 1 / 12 cos 3 x + k

 
3.∫ sin 3 x sin 4 x d x = ½ ∫ 2 sin 3 x sin 4 x d x

  = ½ ∫ (cos x – cos 7 x) d x

  = ½ [∫ cos x dx - ∫ cos 7 x dx]

  = ½ [sin x – sin 7x / 7] + k
 

 

 
Solving Calculus ProblemsSolve Calculus ProblemsRange and DomainProperties of Logarithms
Properties of Logarithmic FunctionsProbability FormulasProbability QuestionsProbability Equations

View All Sub topics

HAVE A QUESTION? Chat With Our Tutoring Experts Now