ملحق:قائمة تكاملات الدوال المثلثية

تكاملات مثلثية تحتوي فقط على الجيب (جا)

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

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

\int \sin a_{1} x \sin a_{2} x d x = \frac{\sin [(a_{1} - a_{2}) x]}{2 (a_{1} - a_{2})} - \frac{\sin [(a_{1} + a_{2}) x]}{2 (a_{1} + a_{2})} + C (for | a_{1} | \neq | a_{2} |)

\int \sin^{n} a x d x = - \frac{\sin^{n - 1} a x \cos a x}{n a} + \frac{n - 1}{n} \int \sin^{n - 2} a x d x (for n > 0)

\int \frac{d x}{\sin a x} = \frac{1}{a} \ln | \tan \frac{a x}{2} | + C

\int \frac{d x}{\sin^{n} a x} = \frac{\cos a x}{a (1 - n) \sin^{n - 1} a x} + \frac{n - 2}{n - 1} \int \frac{d x}{\sin^{n - 2} a x} (for n > 1)

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

\int x^{n} \sin a x d x = - \frac{x^{n}}{a} \cos a x + \frac{n}{a} \int x^{n - 1} \cos a x d x (for n > 0)

\int_{\frac{- a}{2}}^{\frac{a}{2}} x^{2} \sin^{2} \frac{n π x}{a} d x = \frac{a^{3} (n^{2} π^{2} - 6)}{24 n^{2} π^{2}} (for n = 2, 4, 6 . . .)

\int \frac{\sin a x}{x} d x = \sum_{n = 0}^{\infty} (- 1)^{n} \frac{(a x)^{2 n + 1}}{(2 n + 1) \cdot (2 n + 1)!} + C

\int \frac{\sin a x}{x^{n}} d x = - \frac{\sin a x}{(n - 1) x^{n - 1}} + \frac{a}{n - 1} \int \frac{\cos a x}{x^{n - 1}} d x

\int \frac{d x}{1 \pm \sin a x} = \frac{1}{a} \tan (\frac{a x}{2} \mp \frac{π}{4}) + C

\int \frac{x d x}{1 + \sin a x} = \frac{x}{a} \tan (\frac{a x}{2} - \frac{π}{4}) + \frac{2}{a^{2}} \ln | \cos (\frac{a x}{2} - \frac{π}{4}) | + C

\int \frac{x d x}{1 - \sin a x} = \frac{x}{a} \cot (\frac{π}{4} - \frac{a x}{2}) + \frac{2}{a^{2}} \ln | \sin (\frac{π}{4} - \frac{a x}{2}) | + C

\int \frac{\sin a x d x}{1 \pm \sin a x} = \pm x + \frac{1}{a} \tan (\frac{π}{4} \mp \frac{a x}{2}) + C

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