Integration Formulas for Trigonometric Functions

Indefinite Integrals of Basic Trig Functions

• Indefinite Integral of sin(x):
  [ \int \sin x ,dx = -\cos x + C ]
• Indefinite Integral of cos(x):
  [ \int \cos x ,dx = \sin x + C ]

Indefinite Integrals of Trig Functions with Linear Arguments

• Indefinite Integral of sin(ax + b):
  [ \int \sin(ax + b) ,dx = -\frac{1}{a} \cos(ax + b) + C ]
• Indefinite Integral of cos(ax + b):
  [ \int \cos(ax + b) ,dx = \frac{1}{a} \sin(ax + b) + C ]

Other Trig Functions

• Indefinite Integral of tan(x):
  [ \int \tan x ,dx = \ln |\sec x| + C ]
• Indefinite Integral of cot(x):
  [ \int \cot x ,dx = \ln |\sin x| + C ]
  • Alternative:
    [ \int \cot x ,dx = -\ln |\csc x| + C ]
• Indefinite Integral of sec(x):
  [ \int \sec x ,dx = \ln |\sec x + \tan x| + C ]
• Indefinite Integral of csc(x):
  [ \int \csc x ,dx = -\ln |\csc x + \cot x| + C ]

Indefinite Integrals of Squared Trig Functions

• Indefinite Integral of sin²(x):
  [ \int \sin^2 x ,dx = \frac{1}{2} x - \frac{1}{4} \sin(2x) + C ]
• Indefinite Integral of sin(2x):
  [ \int \sin(2x) ,dx = -\frac{1}{2} \cos(2x) + C ]
• Indefinite Integral of tan²(x):
  [ \int \tan^2 x ,dx = \tan x - x + C ]
• Indefinite Integral of cot²(x):
  [ \int \cot^2 x ,dx = -\cot x - x + C ]
• Indefinite Integral of sec²(x):
  [ \int \sec^2 x ,dx = \tan x + C ]
• Indefinite Integral of csc²(x):
  [ \int \csc^2 x ,dx = -\cot x + C ]

Reduction Formulas

• Integral of sinⁿ(x):
  [ \int \sin^n x ,dx = -\frac{1}{n} \sin^{n-1} x \cos x + \frac{n-1}{n} \int \sin^{n-2} x ,dx ]
• Integral of cosⁿ(x):
  [ \int \cos^n x ,dx = \frac{1}{n} \cos^{n-1} x \sin x + \frac{n-1}{n} \int \cos^{n-2} x ,dx ]
• Integral of tanⁿ(x):
  [ \int \tan^n x ,dx = \frac{1}{n-1} \tan^{n-1} x - \int \tan^{n-2} x ,dx ]
  • Note: n cannot equal 1.

Additional Resources

• Formula sheets for trig and integration formulas are available in the description for review.