Интегралы от тригонометрических функций

Неопределённые интегралы от тригонометрических функций, их степеней и произведений.

\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 t g a x d x = - \frac{1}{a} ln cos a x + C

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

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

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

\int \frac{1}{cos x} d x = ln ∣ \frac{1}{cos x} + t g x ∣ = 2 arth (t g \frac{x}{2}) + C

\int \frac{1}{cos ^{2} a x} d x = \frac{1}{a} t g a x + C

\int \frac{d x}{cos ^{3} x} = \frac{t g x}{2 cos x} + \frac{1}{2} ln \frac{1}{cos x} + t g x + C

\int \frac{t g x}{cos x} d x = \frac{1}{cos x} + C

\int \frac{t g x}{cos ^{2} x} d x = \frac{1}{2 cos ^{2} x} + C

\int \frac{t g x}{cos ^{n} x} d x = \frac{1}{n cos ^{n} x} + C, n \neq = 0

\int \frac{1}{sin x} d x = ln t g \frac{x}{2} = ln ∣ \frac{1}{sin x} - ct g x ∣ + C

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

\int \frac{d x}{sin ^{3} x} = - \frac{ct g x}{2 sin x} + \frac{1}{2} ln \frac{1}{sin x} - ct g x + C

\int \frac{1}{sin ^{n} x} ct g x d x = - \frac{1}{n sin ^{n} x} + C, n \neq = 0

\int \frac{d x}{sin x cos x} = ln ∣ t g x ∣ + C