Indefinite Integrals of Trigonometric Functions Notes

Indefinite Integrals of Trigonometric Functions

Basic Trigonometric Derivatives and Corresponding Indefinite Integrals

• Sine and Cosine

• Derivative:

  • ( \frac{d}{dx} (\sin x) = \cos x )
  • ( \frac{d}{dx} (\cos x) = -\sin x )

• Indefinite Integrals:

  • ( \int \cos x \, dx = \sin x + C )
  • ( \int -\sin x \, dx = \cos x + C )

• Tangent and Secant

• Derivative:

  • ( \frac{d}{dx} (\tan x) = \sec^2 x )
  • ( \frac{d}{dx} (\sec x) = \sec x \tan x )

• Indefinite Integrals:

  • ( \int \sec^2 x \, dx = \tan x + C )
  • ( \int \sec x \tan x \, dx = \sec x + C )

• Cosecant and Cotangent

• Derivative:

  • ( \frac{d}{dx} (\csc x) = -\csc x \cot x )
  • ( \frac{d}{dx} (\cot x) = -\csc^2 x )

• Indefinite Integrals:

  • ( \int -\csc x \cot x \, dx = \csc x + C )
  • ( \int -\csc^2 x \, dx = \cot x + C )

Example Problems

Example No. 1

• Given:
  ( \int (5\sin x - 2\cos x) \, dx )
• Solution:
• Break down into two integrals:
  • ( \int 5\sin x \, dx - \int 2\cos x \, dx )
• Evaluate:
  • ( 5(-\cos x) + C1 - 2(\sin x) + C2 )
• Combine constants:
  • Result:
    ( -5\cos x - 2\sin x + C )

Example No. 2

• Given:
  ( \int (\sec^2 x - \csc^2 x) \, dx )
• Solution:
• Integrate each term:
  • ( \int \sec^2 x \, dx + \int -\csc^2 x \, dx )
• Evaluate:
  • ( \tan x + C1 + \cot x + C2 )
• Combine constants:
  • Result:
    ( \tan x + \cot x + C )

Example No. 3

• Given:
  ( \int -4\sin x \, dx )
• Solution:
• Evaluate:
  • ( -4(-\cos x) + C )
• Simplified Result:
  ( 4\cos x + C )

Example No. 4

• Given:
  ( \int 8\sec^2 x \, dx )
• Solution:
• Evaluate:
  • ( 8\tan x + C )

Example No. 5

• Given:
  ( \int -5\csc x \cot x \, dx )
• Suggestion:
• Utilize known derivative relationship for cosecant and cotangent.