Integrals with Trigonometric Functions

Indefinite integrals of trigonometric functions, their powers and their products.

\int sin a x d x = - \frac{1}{a} cos a x + C

\int sin^{2} a x d x = \frac{x}{2} - \frac{sin 2 a x}{4 a} + C

\int sin^{3} a x d x = - \frac{3 cos a x}{4 a} + \frac{cos 3 a x}{12 a} + C

\int sin^{n} a x d x = - \frac{1}{a} cos a x_{2} F_{1} [\frac{1}{2}, \frac{1 - n}{2}, \frac{3}{2}, cos^{2} a x] + C

\int cos a x d x = \frac{1}{a} sin a x + C

\int cos^{2} a x d x = \frac{x}{2} + \frac{sin 2 a x}{4 a} + C

\int cos^{3} a x d x = \frac{3 sin a x}{4 a} + \frac{sin 3 a x}{12 a} + C

\int cos^{p} a x d x = - \frac{1}{a ( 1 + p )} cos^{1 + p} a x \times_{2} F_{1} [\frac{1 + p}{2}, \frac{1}{2}, \frac{3 + p}{2}, cos^{2} a x] + C

\int cos x sin x d x = \frac{1}{2} sin^{2} x + c_{1} = - \frac{1}{2} cos^{2} x + c_{2} = - \frac{1}{4} cos 2 x + c_{3} + C

\int cos a x sin b x d x = \frac{cos [( a - b ) x ]}{2 ( a - b )} - \frac{cos [( a + b ) x ]}{2 ( a + b )} + C, a \neq = b

\int sin^{2} a x cos b x d x = - \frac{sin [( 2 a - b ) x ]}{4 ( 2 a - b )} + \frac{sin b x}{2 b} - \frac{sin [( 2 a + b ) x ]}{4 ( 2 a + b )} + C

\int sin^{2} x cos x d x = \frac{1}{3} sin^{3} x + C

\int cos^{2} a x sin b x d x = \frac{cos [( 2 a - b ) x ]}{4 ( 2 a - b )} - \frac{cos b x}{2 b} - \frac{cos [( 2 a + b ) x ]}{4 ( 2 a + b )} + C

\int cos^{2} a x sin a x d x = - \frac{1}{3 a} cos^{3} a x + C

\int sin^{2} a x cos^{2} b x d x = \frac{x}{4} - \frac{sin 2 a x}{8 a} - \frac{sin [ 2 ( a - b ) x ]}{16 ( a - b )} + \frac{sin 2 b x}{8 b} - \frac{sin [ 2 ( a + b ) x ]}{16 ( a + b )} + C

\int sin^{2} a x cos^{2} a x d x = \frac{x}{8} - \frac{sin 4 a x}{32 a} + C

\int tan a x d x = - \frac{1}{a} ln cos a x + C

\int tan^{2} a x d x = - x + \frac{1}{a} tan a x + C

\int tan^{n} a x d x = \frac{tan ^{n + 1} a x}{a ( 1 + n )} \times_{2} F_{1} (\frac{n + 1}{2}, 1, \frac{n + 3}{2}, - tan^{2} a x) + C

\int tan^{3} a x d x = \frac{1}{a} ln cos a x + \frac{1}{2 a} sec^{2} a x + C

\int sec x d x = ln ∣ sec x + tan x ∣ = 2 tanh^{- 1} (tan \frac{x}{2}) + C

\int sec^{2} a x d x = \frac{1}{a} tan a x + C

\int sec^{3} x d x = \frac{1}{2} sec x tan x + \frac{1}{2} ln ∣ sec x + tan x ∣ + C

\int sec x tan x d x = sec x + C

\int sec^{2} x tan x d x = \frac{1}{2} sec^{2} x + C

\int sec^{n} x tan x d x = \frac{1}{n} sec^{n} x + C, n \neq = 0

\int csc x d x = ln tan \frac{x}{2} = ln ∣ csc x - cot x ∣ + C

\int csc^{2} a x d x = - \frac{1}{a} cot a x + C

\int csc^{3} x d x = - \frac{1}{2} cot x csc x + \frac{1}{2} ln ∣ csc x - cot x ∣ + C

\int csc^{n} x cot x d x = - \frac{1}{n} csc^{n} x + C, n \neq = 0

\int sec x csc x d x = ln ∣ tan x ∣ + C

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