त्रिकोणमितीय फलनों के अभिन्नों की सूची
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{{Trigonometry}}
{{Trigonometry}}
निम्नलिखित [[त्रिकोणमितीय कार्य]]ों के [[अभिन्न]] अंग ([[ antiderivative ]] [[फ़ंक्शन (गणित)]]) की एक सूची है। घातांकीय और त्रिकोणमितीय दोनों कार्यों से जुड़े प्रतिअवकलन के लिए, घातांकीय कार्यों के [[अभिन्नों की सूची]] देखें। प्रतिअवकलन कार्यों की पूरी सूची के लिए, अभिन्नों की सूचियाँ देखें। त्रिकोणमितीय कार्यों से जुड़े विशेष प्रतिअवकलन के लिए, [[त्रिकोणमितीय अभिन्न]] अंग देखें।
निम्नलिखित [[त्रिकोणमितीय कार्य]]ों के [[अभिन्न]] अंग ([[ antiderivative ]] [[फ़ंक्शन (गणित)]]) की एक सूची है। घातांकीय और त्रिकोणमितीय दोनों कार्यों से जुड़े प्रतिअवकलन के लिए, घातांकीय कार्यों के [[अभिन्नों की सूची]] देखें। प्रतिअवकलन कार्यों की पूरी सूची के लिए, अभिन्नों की सूचियाँ देखें। त्रिकोणमितीय कार्यों से जुड़े विशेष प्रतिअवकलन के लिए, [[त्रिकोणमितीय अभिन्न]] अंग देखें।
Revision as of 18:07, 20 July 2023
निम्नलिखित त्रिकोणमितीय कार्य ों के अभिन्न अंग (antiderivative फ़ंक्शन (गणित) ) की एक सूची है। घातांकीय और त्रिकोणमितीय दोनों कार्यों से जुड़े प्रतिअवकलन के लिए, घातांकीय कार्यों के अभिन्नों की सूची देखें। प्रतिअवकलन कार्यों की पूरी सूची के लिए, अभिन्नों की सूचियाँ देखें। त्रिकोणमितीय कार्यों से जुड़े विशेष प्रतिअवकलन के लिए, त्रिकोणमितीय अभिन्न अंग देखें।
आम तौर पर, यदि फ़ंक्शन sin x {\displaystyle \sin x} cos x {\displaystyle \cos x}
∫ a cos n x d x = a n sin n x + C {\displaystyle \int a\cos nx\,dx={\frac {a}{n}}\sin nx+C}
सभी सूत्रों में स्थिरांक a को शून्येतर माना जाता है, और C एकीकरण के स्थिरांक को दर्शाता है।
== इंटीग्रैंड्स में केवल उन लोगों के == शामिल है
∫ sin a x d x = − 1 a cos a x + C {\displaystyle \int \sin ax\,dx=-{\frac {1}{a}}\cos ax+C} ∫ sin 2 a x d x = x 2 − 1 4 a sin 2 a x + C = x 2 − 1 2 a sin a x cos a x + C {\displaystyle \int \sin ^{2}{ax}\,dx={\frac {x}{2}}-{\frac {1}{4a}}\sin 2ax+C={\frac {x}{2}}-{\frac {1}{2a}}\sin ax\cos ax+C} ∫ sin 3 a x d x = cos 3 a x 12 a − 3 cos a x 4 a + C {\displaystyle \int \sin ^{3}{ax}\,dx={\frac {\cos 3ax}{12a}}-{\frac {3\cos ax}{4a}}+C} ∫ x sin 2 a x d x = x 2 4 − x 4 a sin 2 a x − 1 8 a 2 cos 2 a x + C {\displaystyle \int x\sin ^{2}{ax}\,dx={\frac {x^{2}}{4}}-{\frac {x}{4a}}\sin 2ax-{\frac {1}{8a^{2}}}\cos 2ax+C} ∫ x 2 sin 2 a x d x = x 3 6 − ( x 2 4 a − 1 8 a 3 ) sin 2 a x − x 4 a 2 cos 2 a x + C {\displaystyle \int x^{2}\sin ^{2}{ax}\,dx={\frac {x^{3}}{6}}-\left({\frac {x^{2}}{4a}}-{\frac {1}{8a^{3}}}\right)\sin 2ax-{\frac {x}{4a^{2}}}\cos 2ax+C} ∫ x sin a x d x = sin a x a 2 − x cos a x a + C {\displaystyle \int x\sin ax\,dx={\frac {\sin ax}{a^{2}}}-{\frac {x\cos ax}{a}}+C} ∫ ( sin b 1 x ) ( sin b 2 x ) d x = sin ( ( b 2 − b 1 ) x ) 2 ( b 2 − b 1 ) − sin ( ( b 1 + b 2 ) x ) 2 ( b 1 + b 2 ) + C (for | b 1 | ≠ | b 2 | ) {\displaystyle \int (\sin b_{1}x)(\sin b_{2}x)\,dx={\frac {\sin((b_{2}-b_{1})x)}{2(b_{2}-b_{1})}}-{\frac {\sin((b_{1}+b_{2})x)}{2(b_{1}+b_{2})}}+C\qquad {\mbox{(for }}|b_{1}|\neq |b_{2}|{\mbox{)}}} ∫ sin n a x d x = − sin n − 1 a x cos a x n a + n − 1 n ∫ sin n − 2 a x d x (for n > 0 ) {\displaystyle \int \sin ^{n}{ax}\,dx=-{\frac {\sin ^{n-1}ax\cos ax}{na}}+{\frac {n-1}{n}}\int \sin ^{n-2}ax\,dx\qquad {\mbox{(for }}n>0{\mbox{)}}} ∫ d x sin a x = − 1 a ln | csc a x + cot a x | + C {\displaystyle \int {\frac {dx}{\sin ax}}=-{\frac {1}{a}}\ln {\left|\csc {ax}+\cot {ax}\right|}+C} ∫ d x sin n a x = cos a x a ( 1 − n ) sin n − 1 a x + n − 2 n − 1 ∫ d x sin n − 2 a x (for n > 1 ) {\displaystyle \int {\frac {dx}{\sin ^{n}ax}}={\frac {\cos ax}{a(1-n)\sin ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\sin ^{n-2}ax}}\qquad {\mbox{(for }}n>1{\mbox{)}}} ∫ x n sin a x d x = − x n a cos a x + n a ∫ x n − 1 cos a x d x = ∑ k = 0 2 k ≤ n ( − 1 ) k + 1 x n − 2 k a 1 + 2 k n ! ( n − 2 k ) ! cos a x + ∑ k = 0 2 k + 1 ≤ n ( − 1 ) k x n − 1 − 2 k a 2 + 2 k n ! ( n − 2 k − 1 ) ! sin a x = − ∑ k = 0 n x n − k a 1 + k n ! ( n − k ) ! cos ( a x + k π 2 ) (for n > 0 ) {\displaystyle {\begin{aligned}\int x^{n}\sin ax\,dx&=-{\frac {x^{n}}{a}}\cos ax+{\frac {n}{a}}\int x^{n-1}\cos ax\,dx\\&=\sum _{k=0}^{2k\leq n}(-1)^{k+1}{\frac {x^{n-2k}}{a^{1+2k}}}{\frac {n!}{(n-2k)!}}\cos ax+\sum _{k=0}^{2k+1\leq n}(-1)^{k}{\frac {x^{n-1-2k}}{a^{2+2k}}}{\frac {n!}{(n-2k-1)!}}\sin ax\\&=-\sum _{k=0}^{n}{\frac {x^{n-k}}{a^{1+k}}}{\frac {n!}{(n-k)!}}\cos \left(ax+k{\frac {\pi }{2}}\right)\qquad {\mbox{(for }}n>0{\mbox{)}}\end{aligned}}} ∫ sin a x x d x = ∑ n = 0 ∞ ( − 1 ) n ( a x ) 2 n + 1 ( 2 n + 1 ) ⋅ ( 2 n + 1 ) ! + C {\displaystyle \int {\frac {\sin ax}{x}}\,dx=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(ax)^{2n+1}}{(2n+1)\cdot (2n+1)!}}+C} ∫ sin a x x n d x = − sin a x ( n − 1 ) x n − 1 + a n − 1 ∫ cos a x x n − 1 d x {\displaystyle \int {\frac {\sin ax}{x^{n}}}\,dx=-{\frac {\sin ax}{(n-1)x^{n-1}}}+{\frac {a}{n-1}}\int {\frac {\cos ax}{x^{n-1}}}\,dx} ∫ sin ( a x 2 + b x + c ) d x = { a π 2 cos ( b 2 − 4 a c 4 a ) S ( 2 a x + b 2 a π ) + a π 2 sin ( b 2 − 4 a c 4 a ) C ( 2 a x + b 2 a π ) t o b 2 − 4 a c > 0 a π 2 cos ( b 2 − 4 a c 4 a ) S ( 2 a x + b 2 a π ) − a π 2 sin ( b 2 − 4 a c 4 a ) C ( 2 a x + b 2 a π ) t o b 2 − 4 a c < 0 f o r a ╱ = ∫ d x 1 ± sin a x = 1 a tan ( a x 2 ∓ π 4 ) + C {\displaystyle \int {\frac {dx}{1\pm \sin ax}}={\frac {1}{a}}\tan \left({\frac {ax}{2}}\mp {\frac {\pi }{4}}\right)+C} ∫ x d x 1 + sin a x = x a tan ( a x 2 − π 4 ) + 2 a 2 ln | cos ( a x 2 − π 4 ) | + C {\displaystyle \int {\frac {x\,dx}{1+\sin ax}}={\frac {x}{a}}\tan \left({\frac {ax}{2}}-{\frac {\pi }{4}}\right)+{\frac {2}{a^{2}}}\ln \left|\cos \left({\frac {ax}{2}}-{\frac {\pi }{4}}\right)\right|+C} ∫ x d x 1 − sin a x = x a cot ( π 4 − a x 2 ) + 2 a 2 ln | sin ( π 4 − a x 2 ) | + C {\displaystyle \int {\frac {x\,dx}{1-\sin ax}}={\frac {x}{a}}\cot \left({\frac {\pi }{4}}-{\frac {ax}{2}}\right)+{\frac {2}{a^{2}}}\ln \left|\sin \left({\frac {\pi }{4}}-{\frac {ax}{2}}\right)\right|+C} ∫ sin a x d x 1 ± sin a x = ± x + 1 a tan ( π 4 ∓ a x 2 ) + C {\displaystyle \int {\frac {\sin ax\,dx}{1\pm \sin ax}}=\pm x+{\frac {1}{a}}\tan \left({\frac {\pi }{4}}\mp {\frac {ax}{2}}\right)+C}
इंटीग्रैंड्स में केवल कोज्या शामिल है
∫ cos a x d x = 1 a sin a x + C {\displaystyle \int \cos ax\,dx={\frac {1}{a}}\sin ax+C} ∫ cos 2 a x d x = x 2 + 1 4 a sin 2 a x + C = x 2 + 1 2 a sin a x cos a x + C {\displaystyle \int \cos ^{2}{ax}\,dx={\frac {x}{2}}+{\frac {1}{4a}}\sin 2ax+C={\frac {x}{2}}+{\frac {1}{2a}}\sin ax\cos ax+C} ∫ cos n a x d x = cos n − 1 a x sin a x n a + n − 1 n ∫ cos n − 2 a x d x (for n > 0 ) {\displaystyle \int \cos ^{n}ax\,dx={\frac {\cos ^{n-1}ax\sin ax}{na}}+{\frac {n-1}{n}}\int \cos ^{n-2}ax\,dx\qquad {\mbox{(for }}n>0{\mbox{)}}} ∫ x cos a x d x = cos a x a 2 + x sin a x a + C {\displaystyle \int x\cos ax\,dx={\frac {\cos ax}{a^{2}}}+{\frac {x\sin ax}{a}}+C} ∫ x 2 cos 2 a x d x = x 3 6 + ( x 2 4 a − 1 8 a 3 ) sin 2 a x + x 4 a 2 cos 2 a x + C {\displaystyle \int x^{2}\cos ^{2}{ax}\,dx={\frac {x^{3}}{6}}+\left({\frac {x^{2}}{4a}}-{\frac {1}{8a^{3}}}\right)\sin 2ax+{\frac {x}{4a^{2}}}\cos 2ax+C} ∫ x n cos a x d x = x n sin a x a − n a ∫ x n − 1 sin a x d x = ∑ k = 0 2 k + 1 ≤ n ( − 1 ) k x n − 2 k − 1 a 2 + 2 k n ! ( n − 2 k − 1 ) ! cos a x + ∑ k = 0 2 k ≤ n ( − 1 ) k x n − 2 k a 1 + 2 k n ! ( n − 2 k ) ! sin a x = ∑ k = 0 n ( − 1 ) ⌊ k / 2 ⌋ x n − k a 1 + k n ! ( n − k ) ! cos ( a x − ( − 1 ) k + 1 2 π 2 ) = ∑ k = 0 n x n − k a 1 + k ∫ cos a x x d x = ln | a x | + ∑ k = 1 ∞ ( − 1 ) k ( a x ) 2 k 2 k ⋅ ( 2 k ) ! + C {\displaystyle \int {\frac {\cos ax}{x}}\,dx=\ln |ax|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(ax)^{2k}}{2k\cdot (2k)!}}+C} ∫ cos a x x n d x = − cos a x ( n − 1 ) x n − 1 − a n − 1 ∫ sin a x x n − 1 d x (for n ≠ 1 ) {\displaystyle \int {\frac {\cos ax}{x^{n}}}\,dx=-{\frac {\cos ax}{(n-1)x^{n-1}}}-{\frac {a}{n-1}}\int {\frac {\sin ax}{x^{n-1}}}\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}} ∫ d x cos a x = 1 a ln | tan ( a x 2 + π 4 ) | + C {\displaystyle \int {\frac {dx}{\cos ax}}={\frac {1}{a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C} ∫ d x cos n a x = sin a x a ( n − 1 ) cos n − 1 a x + n − 2 n − 1 ∫ d x cos n − 2 a x (for n > 1 ) {\displaystyle \int {\frac {dx}{\cos ^{n}ax}}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cos ^{n-2}ax}}\qquad {\mbox{(for }}n>1{\mbox{)}}} ∫ d x 1 + cos a x = 1 a tan a x 2 + C {\displaystyle \int {\frac {dx}{1+\cos ax}}={\frac {1}{a}}\tan {\frac {ax}{2}}+C} ∫ d x 1 − cos a x = − 1 a cot a x 2 + C {\displaystyle \int {\frac {dx}{1-\cos ax}}=-{\frac {1}{a}}\cot {\frac {ax}{2}}+C} ∫ x d x 1 + cos a x = x a tan a x 2 + 2 a 2 ln | cos a x 2 | + C {\displaystyle \int {\frac {x\,dx}{1+\cos ax}}={\frac {x}{a}}\tan {\frac {ax}{2}}+{\frac {2}{a^{2}}}\ln \left|\cos {\frac {ax}{2}}\right|+C} ∫ x d x 1 − cos a x = − x a cot a x 2 + 2 a 2 ln | sin a x 2 | + C {\displaystyle \int {\frac {x\,dx}{1-\cos ax}}=-{\frac {x}{a}}\cot {\frac {ax}{2}}+{\frac {2}{a^{2}}}\ln \left|\sin {\frac {ax}{2}}\right|+C} ∫ cos a x d x 1 + cos a x = x − 1 a tan a x 2 + C {\displaystyle \int {\frac {\cos ax\,dx}{1+\cos ax}}=x-{\frac {1}{a}}\tan {\frac {ax}{2}}+C} ∫ cos a x d x 1 − cos a x = − x − 1 a cot a x 2 + C {\displaystyle \int {\frac {\cos ax\,dx}{1-\cos ax}}=-x-{\frac {1}{a}}\cot {\frac {ax}{2}}+C} ∫ ( cos a 1 x ) ( cos a 2 x ) d x = sin ( ( a 2 − a 1 ) x ) 2 ( a 2 − a 1 ) + sin ( ( a 2 + a 1 ) x ) 2 ( a 2 + a 1 ) + C (for | a 1 | ≠ | a 2 | ) {\displaystyle \int (\cos a_{1}x)(\cos a_{2}x)\,dx={\frac {\sin((a_{2}-a_{1})x)}{2(a_{2}-a_{1})}}+{\frac {\sin((a_{2}+a_{1})x)}{2(a_{2}+a_{1})}}+C\qquad {\mbox{(for }}|a_{1}|\neq |a_{2}|{\mbox{)}}}
∫ tan a x d x = − 1 a ln | cos a x | + C = 1 a ln | sec a x | + C {\displaystyle \int \tan ax\,dx=-{\frac {1}{a}}\ln |\cos ax|+C={\frac {1}{a}}\ln |\sec ax|+C} ∫ tan 2 x d x = tan x − x + C {\displaystyle \int \tan ^{2}{x}\,dx=\tan {x}-x+C} ∫ tan n a x d x = 1 a ( n − 1 ) tan n − 1 a x − ∫ tan n − 2 a x d x (for n ≠ 1 ) {\displaystyle \int \tan ^{n}ax\,dx={\frac {1}{a(n-1)}}\tan ^{n-1}ax-\int \tan ^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}} ∫ d x q tan a x + p = 1 p 2 + q 2 ( p x + q a ln | q sin a x + p cos a x | ) + C (for p 2 + q 2 ≠ 0 ) {\displaystyle \int {\frac {dx}{q\tan ax+p}}={\frac {1}{p^{2}+q^{2}}}(px+{\frac {q}{a}}\ln |q\sin ax+p\cos ax|)+C\qquad {\mbox{(for }}p^{2}+q^{2}\neq 0{\mbox{)}}} ∫ d x tan a x ± 1 = ± x 2 + 1 2 a ln | sin a x ± cos a x | + C {\displaystyle \int {\frac {dx}{\tan ax\pm 1}}=\pm {\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax\pm \cos ax|+C} ∫ tan a x d x tan a x ± 1 = x 2 ∓ 1 2 a ln | sin a x ± cos a x | + C {\displaystyle \int {\frac {\tan ax\,dx}{\tan ax\pm 1}}={\frac {x}{2}}\mp {\frac {1}{2a}}\ln |\sin ax\pm \cos ax|+C}
इंटीग्रैंड्स में केवल सेकेंट (त्रिकोणमितीय फ़ंक्शन) शामिल है सेकेंट फ़ंक्शन का इंटीग्रल देखें।
∫ sec a x d x = 1 a ln | sec a x + tan a x | + C = 1 a ln | tan ( a x 2 + π 4 ) | + C = 1 a artanh ( sin a x ) + C {\displaystyle \int \sec {ax}\,dx={\frac {1}{a}}\ln {\left|\sec {ax}+\tan {ax}\right|}+C={\frac {1}{a}}\ln {\left|\tan {\left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)}\right|}+C={\frac {1}{a}}\operatorname {artanh} {\left(\sin {ax}\right)}+C} ∫ sec 2 x d x = tan x + C {\displaystyle \int \sec ^{2}{x}\,dx=\tan {x}+C} ∫ sec 3 x d x = 1 2 sec x tan x + 1 2 ln | sec x + tan x | + C . {\displaystyle \int \sec ^{3}{x}\,dx={\frac {1}{2}}\sec x\tan x+{\frac {1}{2}}\ln |\sec x+\tan x|+C.} ∫ sec n a x d x = sec n − 2 a x tan a x a ( n − 1 ) + n − 2 n − 1 ∫ sec n − 2 a x d x (for n ≠ 1 ) {\displaystyle \int \sec ^{n}{ax}\,dx={\frac {\sec ^{n-2}{ax}\tan {ax}}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{ax}\,dx\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}} ∫ d x sec x + 1 = x − tan x 2 + C {\displaystyle \int {\frac {dx}{\sec {x}+1}}=x-\tan {\frac {x}{2}}+C} ∫ d x sec x − 1 = − x − cot x 2 + C {\displaystyle \int {\frac {dx}{\sec {x}-1}}=-x-\cot {\frac {x}{2}}+C}
== समाकलन में केवल सहसंयोजक == शामिल है
∫ csc a x d x = − 1 a ln | csc a x + cot a x | + C = 1 a ln | csc a x − cot a x | + C = 1 a ln | tan ( a x 2 ) | + C {\displaystyle \int \csc {ax}\,dx=-{\frac {1}{a}}\ln {\left|\csc {ax}+\cot {ax}\right|}+C={\frac {1}{a}}\ln {\left|\csc {ax}-\cot {ax}\right|}+C={\frac {1}{a}}\ln {\left|\tan {\left({\frac {ax}{2}}\right)}\right|}+C} ∫ csc 2 x d x = − cot x + C {\displaystyle \int \csc ^{2}{x}\,dx=-\cot {x}+C} ∫ csc 3 x d x = − 1 2 csc x cot x − 1 2 ln | csc x + cot x | + C = − 1 2 csc x cot x + 1 2 ln | csc x − cot x | + C {\displaystyle \int \csc ^{3}{x}\,dx=-{\frac {1}{2}}\csc x\cot x-{\frac {1}{2}}\ln |\csc x+\cot x|+C=-{\frac {1}{2}}\csc x\cot x+{\frac {1}{2}}\ln |\csc x-\cot x|+C} ∫ csc n a x d x = − csc n − 2 a x cot a x a ( n − 1 ) + n − 2 n − 1 ∫ csc n − 2 a x d x (for n ≠ 1 ) {\displaystyle \int \csc ^{n}{ax}\,dx=-{\frac {\csc ^{n-2}{ax}\cot {ax}}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \csc ^{n-2}{ax}\,dx\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}} ∫ d x csc x + 1 = x − 2 cot x 2 + 1 + C {\displaystyle \int {\frac {dx}{\csc {x}+1}}=x-{\frac {2}{\cot {\frac {x}{2}}+1}}+C} ∫ d x csc x − 1 = − x + 2 cot x 2 − 1 + C {\displaystyle \int {\frac {dx}{\csc {x}-1}}=-x+{\frac {2}{\cot {\frac {x}{2}}-1}}+C}
समाकलन में केवल कोटैंजेंट शामिल है
∫ cot a x d x = 1 a ln | sin a x | + C {\displaystyle \int \cot ax\,dx={\frac {1}{a}}\ln |\sin ax|+C} ∫ cot 2 x d x = − cot x − x + C {\displaystyle \int \cot ^{2}{x}\,dx=-\cot {x}-x+C} ∫ cot n a x d x = − 1 a ( n − 1 ) cot n − 1 a x − ∫ cot n − 2 a x d x (for n ≠ 1 ) {\displaystyle \int \cot ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\cot ^{n-1}ax-\int \cot ^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}} ∫ d x 1 + cot a x = ∫ tan a x d x tan a x + 1 = x 2 − 1 2 a ln | sin a x + cos a x | + C {\displaystyle \int {\frac {dx}{1+\cot ax}}=\int {\frac {\tan ax\,dx}{\tan ax+1}}={\frac {x}{2}}-{\frac {1}{2a}}\ln |\sin ax+\cos ax|+C} ∫ d x 1 − cot a x = ∫ tan a x d x tan a x − 1 = x 2 + 1 2 a ln | sin a x − cos a x | + C {\displaystyle \int {\frac {dx}{1-\cot ax}}=\int {\frac {\tan ax\,dx}{\tan ax-1}}={\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax-\cos ax|+C}
साइन और कोसाइन दोनों को शामिल करने वाला समाकलन एक अभिन्न अंग जो साइन और कोसाइन का एक तर्कसंगत कार्य है, उसका मूल्यांकन बायोचे के नियमों का उपयोग करके किया जा सकता है।
∫ d x cos a x ± sin a x = 1 a 2 ln | tan ( a x 2 ± π 8 ) | + C {\displaystyle \int {\frac {dx}{\cos ax\pm \sin ax}}={\frac {1}{a{\sqrt {2}}}}\ln \left|\tan \left({\frac {ax}{2}}\pm {\frac {\pi }{8}}\right)\right|+C} ∫ d x ( cos a x ± sin a x ) 2 = 1 2 a tan ( a x ∓ π 4 ) + C {\displaystyle \int {\frac {dx}{(\cos ax\pm \sin ax)^{2}}}={\frac {1}{2a}}\tan \left(ax\mp {\frac {\pi }{4}}\right)+C} ∫ d x ( cos x + sin x ) n = 1 2 ( n − 1 ) ( sin x − cos x ( cos x + sin x ) n − 1 + ( n − 2 ) ∫ d x ( cos x + sin x ) n − 2 ) {\displaystyle \int {\frac {dx}{(\cos x+\sin x)^{n}}}={\frac {1}{2(n-1)}}\left({\frac {\sin x-\cos x}{(\cos x+\sin x)^{n-1}}}+(n-2)\int {\frac {dx}{(\cos x+\sin x)^{n-2}}}\right)} ∫ cos a x d x cos a x + sin a x = x 2 + 1 2 a ln | sin a x + cos a x | + C {\displaystyle \int {\frac {\cos ax\,dx}{\cos ax+\sin ax}}={\frac {x}{2}}+{\frac {1}{2a}}\ln \left|\sin ax+\cos ax\right|+C} ∫ cos a x d x cos a x − sin a x = x 2 − 1 2 a ln | sin a x − cos a x | + C {\displaystyle \int {\frac {\cos ax\,dx}{\cos ax-\sin ax}}={\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax-\cos ax\right|+C} ∫ sin a x d x cos a x + sin a x = x 2 − 1 2 a ln | sin a x + cos a x | + C {\displaystyle \int {\frac {\sin ax\,dx}{\cos ax+\sin ax}}={\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax+\cos ax\right|+C} ∫ sin a x d x cos a x − sin a x = − x 2 − 1 2 a ln | sin a x − cos a x | + C {\displaystyle \int {\frac {\sin ax\,dx}{\cos ax-\sin ax}}=-{\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax-\cos ax\right|+C} ∫ cos a x d x ( sin a x ) ( 1 + cos a x ) = − 1 4 a tan 2 a x 2 + 1 2 a ln | tan a x 2 | + C {\displaystyle \int {\frac {\cos ax\,dx}{(\sin ax)(1+\cos ax)}}=-{\frac {1}{4a}}\tan ^{2}{\frac {ax}{2}}+{\frac {1}{2a}}\ln \left|\tan {\frac {ax}{2}}\right|+C} ∫ cos a x d x ( sin a x ) ( 1 − cos a x ) = − 1 4 a cot 2 a x 2 − 1 2 a ln | tan a x 2 | + C {\displaystyle \int {\frac {\cos ax\,dx}{(\sin ax)(1-\cos ax)}}=-{\frac {1}{4a}}\cot ^{2}{\frac {ax}{2}}-{\frac {1}{2a}}\ln \left|\tan {\frac {ax}{2}}\right|+C} ∫ sin a x d x ( cos a x ) ( 1 + sin a x ) = 1 4 a cot 2 ( a x 2 + π 4 ) + 1 2 a ln | tan ( a x 2 + π 4 ) | + C {\displaystyle \int {\frac {\sin ax\,dx}{(\cos ax)(1+\sin ax)}}={\frac {1}{4a}}\cot ^{2}\left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)+{\frac {1}{2a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C} ∫ sin a x d x ( cos a x ) ( 1 − sin a x ) = 1 4 a tan 2 ( a x 2 + π 4 ) − 1 2 a ln | tan ( a x 2 + π 4 ) | + C {\displaystyle \int {\frac {\sin ax\,dx}{(\cos ax)(1-\sin ax)}}={\frac {1}{4a}}\tan ^{2}\left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)-{\frac {1}{2a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C} ∫ ( sin a x ) ( cos a x ) d x = 1 2 a sin 2 a x + C {\displaystyle \int (\sin ax)(\cos ax)\,dx={\frac {1}{2a}}\sin ^{2}ax+C} ∫ ( sin a 1 x ) ( cos a 2 x ) d x = − cos ( ( a 1 − a 2 ) x ) 2 ( a 1 − a 2 ) − cos ( ( a 1 + a 2 ) x ) 2 ( a 1 + a 2 ) + C (for | a 1 | ≠ | a 2 | ) {\displaystyle \int (\sin a_{1}x)(\cos a_{2}x)\,dx=-{\frac {\cos((a_{1}-a_{2})x)}{2(a_{1}-a_{2})}}-{\frac {\cos((a_{1}+a_{2})x)}{2(a_{1}+a_{2})}}+C\qquad {\mbox{(for }}|a_{1}|\neq |a_{2}|{\mbox{)}}} ∫ ( sin n a x ) ( cos a x ) d x = 1 a ( n + 1 ) sin n + 1 a x + C (for n ≠ − 1 ) {\displaystyle \int (\sin ^{n}ax)(\cos ax)\,dx={\frac {1}{a(n+1)}}\sin ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}} ∫ ( sin a x ) ( cos n a x ) d x = − 1 a ( n + 1 ) cos n + 1 a x + C (for n ≠ − 1 ) {\displaystyle \int (\sin ax)(\cos ^{n}ax)\,dx=-{\frac {1}{a(n+1)}}\cos ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}} ∫ ( sin n a x ) ( cos m a x ) d x = − ( sin n − 1 a x ) ( cos m + 1 a x ) a ( n + m ) + n − 1 n + m ∫ ( sin n − 2 a x ) ( cos m a x ) d x (for m , n > 0 ) = ( sin n + 1 a x ) ( cos m − 1 a x ) a ( n + m ) + m − 1 n + m ∫ ( sin n a x ) ( cos m − 2 a x ) d x (for m , n > 0 ) {\displaystyle {\begin{aligned}\int (\sin ^{n}ax)(\cos ^{m}ax)\,dx&=-{\frac {(\sin ^{n-1}ax)(\cos ^{m+1}ax)}{a(n+m)}}+{\frac {n-1}{n+m}}\int (\sin ^{n-2}ax)(\cos ^{m}ax)\,dx\qquad {\mbox{(for }}m,n>0{\mbox{)}}\\&={\frac {(\sin ^{n+1}ax)(\cos ^{m-1}ax)}{a(n+m)}}+{\frac {m-1}{n+m}}\int (\sin ^{n}ax)(\cos ^{m-2}ax)\,dx\qquad {\mbox{(for }}m,n>0{\mbox{)}}\end{aligned}}} ∫ d x ( sin a x ) ( cos a x ) = 1 a ln | tan a x | + C {\displaystyle \int {\frac {dx}{(\sin ax)(\cos ax)}}={\frac {1}{a}}\ln \left|\tan ax\right|+C} ∫ d x ( sin a x ) ( cos n a x ) = 1 a ( n − 1 ) cos n − 1 a x + ∫ d x ( sin a x ) ( cos n − 2 a x ) (for n ≠ 1 ) {\displaystyle \int {\frac {dx}{(\sin ax)(\cos ^{n}ax)}}={\frac {1}{a(n-1)\cos ^{n-1}ax}}+\int {\frac {dx}{(\sin ax)(\cos ^{n-2}ax)}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}} ∫ d x ( sin n a x ) ( cos a x ) = − 1 a ( n − 1 ) sin n − 1 a x + ∫ d x ( sin n − 2 a x ) ( cos a x ) (for n ≠ 1 ) {\displaystyle \int {\frac {dx}{(\sin ^{n}ax)(\cos ax)}}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}}+\int {\frac {dx}{(\sin ^{n-2}ax)(\cos ax)}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}} ∫ sin a x d x cos n a x = 1 a ( n − 1 ) cos n − 1 a x + C (for n ≠ 1 ) {\displaystyle \int {\frac {\sin ax\,dx}{\cos ^{n}ax}}={\frac {1}{a(n-1)\cos ^{n-1}ax}}+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}} ∫ sin 2 a x d x cos a x = − 1 a sin a x + 1 a ln | tan ( π 4 + a x 2 ) | + C {\displaystyle \int {\frac {\sin ^{2}ax\,dx}{\cos ax}}=-{\frac {1}{a}}\sin ax+{\frac {1}{a}}\ln \left|\tan \left({\frac {\pi }{4}}+{\frac {ax}{2}}\right)\right|+C} ∫ sin 2 a x d x cos n a x = sin a x a ( n − 1 ) cos n − 1 a x − 1 n − 1 ∫ d x cos n − 2 a x (for n ≠ 1 ) {\displaystyle \int {\frac {\sin ^{2}ax\,dx}{\cos ^{n}ax}}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}}-{\frac {1}{n-1}}\int {\frac {dx}{\cos ^{n-2}ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}} ∫ sin 2 x 1 + cos 2 x d x = 2 arctangant ( tan x 2 ) − x (for x in ] − π 2 ; + π 2 [ ) = 2 arctangant ( tan x 2 ) − arctangant ( tan x ) (this time x being any real number ) {\displaystyle {\begin{aligned}\int {\frac {\sin ^{2}x}{1+\cos ^{2}x}}\,dx&={\sqrt {2}}\operatorname {arctangant} \left({\frac {\tan x}{\sqrt {2}}}\right)-x\qquad {\mbox{(for x in}}]-{\frac {\pi }{2}};+{\frac {\pi }{2}}[{\mbox{)}}\\&={\sqrt {2}}\operatorname {arctangant} \left({\frac {\tan x}{\sqrt {2}}}\right)-\operatorname {arctangant} \left(\tan x\right)\qquad {\mbox{(this time x being any real number }}{\mbox{)}}\end{aligned}}} ∫ sin n a x d x cos a x = − sin n − 1 a x a ( n − 1 ) + ∫ sin n − 2 a x d x cos a x (for n ≠ 1 ) {\displaystyle \int {\frac {\sin ^{n}ax\,dx}{\cos ax}}=-{\frac {\sin ^{n-1}ax}{a(n-1)}}+\int {\frac {\sin ^{n-2}ax\,dx}{\cos ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}} ∫ sin n a x d x cos m a x = { sin n + 1 a x a ( m − 1 ) cos m − 1 a x − n − m + 2 m − 1 ∫ sin n a x d x cos m − 2 a x (for m ≠ 1 ) sin n − 1 a x a ( m − 1 ) cos m − 1 a x − n − 1 m − 1 ∫ sin n − 2 a x d x cos m − 2 a x (for m ≠ 1 ) − sin n − 1 a x a ( n − m ) cos m − 1 a x + n − 1 n − m ∫ sin n − 2 a x d x cos m a x (for m ≠ n ) {\displaystyle \int {\frac {\sin ^{n}ax\,dx}{\cos ^{m}ax}}={\begin{cases}{\frac {\sin ^{n+1}ax}{a(m-1)\cos ^{m-1}ax}}-{\frac {n-m+2}{m-1}}\int {\frac {\sin ^{n}ax\,dx}{\cos ^{m-2}ax}}&{\mbox{(for }}m\neq 1{\mbox{)}}\\{\frac {\sin ^{n-1}ax}{a(m-1)\cos ^{m-1}ax}}-{\frac {n-1}{m-1}}\int {\frac {\sin ^{n-2}ax\,dx}{\cos ^{m-2}ax}}&{\mbox{(for }}m\neq 1{\mbox{)}}\\-{\frac {\sin ^{n-1}ax}{a(n-m)\cos ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\sin ^{n-2}ax\,dx}{\cos ^{m}ax}}&{\mbox{(for }}m\neq n{\mbox{)}}\end{cases}}} ∫ cos a x d x sin n a x = − 1 a ( n − 1 ) sin n − 1 a x + C (for n ≠ 1 ) {\displaystyle \int {\frac {\cos ax\,dx}{\sin ^{n}ax}}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}}+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}} ∫ cos 2 a x d x sin a x = 1 a ( cos a x + ln | tan a x 2 | ) + C {\displaystyle \int {\frac {\cos ^{2}ax\,dx}{\sin ax}}={\frac {1}{a}}\left(\cos ax+\ln \left|\tan {\frac {ax}{2}}\right|\right)+C} ∫ cos 2 a x d x sin n a x = − 1 n − 1 ( cos a x a sin n − 1 a x + ∫ d x sin n − 2 a x ) (for n ≠ 1 ) {\displaystyle \int {\frac {\cos ^{2}ax\,dx}{\sin ^{n}ax}}=-{\frac {1}{n-1}}\left({\frac {\cos ax}{a\sin ^{n-1}ax}}+\int {\frac {dx}{\sin ^{n-2}ax}}\right)\qquad {\mbox{(for }}n\neq 1{\mbox{)}}} ∫ cos n a x d x sin m a x = { − cos n + 1 a x a ( m − 1 ) sin m − 1 a x − n − m + 2 m − 1 ∫ cos n a x d x sin m − 2 a x (for n ≠ 1 ) − cos n − 1 a x a ( m − 1 ) sin m − 1 a x − n − 1 m − 1 ∫ cos n − 2 a x d x sin m − 2 a x (for m ≠ 1 ) cos n − 1 a x a ( n − m ) sin m − 1 a x + n − 1 n − m ∫ cos n − 2 a x d x sin m a x (for m ≠ n ) {\displaystyle \int {\frac {\cos ^{n}ax\,dx}{\sin ^{m}ax}}={\begin{cases}-{\frac {\cos ^{n+1}ax}{a(m-1)\sin ^{m-1}ax}}-{\frac {n-m+2}{m-1}}\int {\frac {\cos ^{n}ax\,dx}{\sin ^{m-2}ax}}&{\mbox{(for }}n\neq 1{\mbox{)}}\\-{\frac {\cos ^{n-1}ax}{a(m-1)\sin ^{m-1}ax}}-{\frac {n-1}{m-1}}\int {\frac {\cos ^{n-2}ax\,dx}{\sin ^{m-2}ax}}&{\mbox{(for }}m\neq 1{\mbox{)}}\\{\frac {\cos ^{n-1}ax}{a(n-m)\sin ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\cos ^{n-2}ax\,dx}{\sin ^{m}ax}}&{\mbox{(for }}m\neq n{\mbox{)}}\end{cases}}}
ज्या और स्पर्शरेखा दोनों को शामिल करने वाला समाकलन
∫ ( sin a x ) ( tan a x ) d x = 1 a ( ln | sec a x + tan a x | − sin a x ) + C {\displaystyle \int (\sin ax)(\tan ax)\,dx={\frac {1}{a}}(\ln |\sec ax+\tan ax|-\sin ax)+C} ∫ tan n a x d x sin 2 a x = 1 a ( n − 1 ) tan n − 1 ( a x ) + C (for n ≠ 1 ) {\displaystyle \int {\frac {\tan ^{n}ax\,dx}{\sin ^{2}ax}}={\frac {1}{a(n-1)}}\tan ^{n-1}(ax)+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
इंटीग्रैंड में कोसाइन और स्पर्शरेखा दोनों शामिल हैं
∫ tan n a x d x cos 2 a x = 1 a ( n + 1 ) tan n + 1 a x + C (for n ≠ − 1 ) {\displaystyle \int {\frac {\tan ^{n}ax\,dx}{\cos ^{2}ax}}={\frac {1}{a(n+1)}}\tan ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}}
इंटीग्रैंड जिसमें साइन और कोटैंजेंट दोनों शामिल हैं
∫ cot n a x d x sin 2 a x = − 1 a ( n + 1 ) cot n + 1 a x + C (for n ≠ − 1 ) {\displaystyle \int {\frac {\cot ^{n}ax\,dx}{\sin ^{2}ax}}=-{\frac {1}{a(n+1)}}\cot ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}}
इंटीग्रैंड में कोसाइन और कोटैंजेंट दोनों शामिल हैं
∫ cot n a x d x cos 2 a x = 1 a ( 1 − n ) tan 1 − n a x + C (for n ≠ 1 ) {\displaystyle \int {\frac {\cot ^{n}ax\,dx}{\cos ^{2}ax}}={\frac {1}{a(1-n)}}\tan ^{1-n}ax+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
समाकलन जिसमें छेदक (त्रिकोणमिति) और स्पर्शरेखा दोनों शामिल हैं
∫ ( sec x ) ( tan x ) d x = sec x + C {\displaystyle \int (\sec x)(\tan x)\,dx=\sec x+C}
समाकलन जिसमें सहसंयोजक और कोटैंजेंट दोनों शामिल हैं
∫ ( csc x ) ( cot x ) d x = − csc x + C {\displaystyle \int (\csc x)(\cot x)\,dx=-\csc x+C}
एक चौथाई अवधि में समाकलन बीटा फ़ंक्शन का उपयोग करना B ( a , b ) {\displaystyle B(a,b)}
∫ 0 π 2 sin n x d x = ∫ 0 π 2 cos n x d x = 1 2 B ( n + 1 2 , 1 2 ) = { n − 1 n ⋅ n − 3 n − 2 ⋯ 3 4 ⋅ 1 2 ⋅ π 2 , if n is even n − 1 n ⋅ n − 3 n − 2 ⋯ 4 5 ⋅ 2 3 , if n is odd and more than 1 1 , if n = 1 {\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}x\,dx={\frac {1}{2}}B\left({\frac {n+1}{2}},{\frac {1}{2}}\right)={\begin{cases}{\frac {n-1}{n}}\cdot {\frac {n-3}{n-2}}\cdots {\frac {3}{4}}\cdot {\frac {1}{2}}\cdot {\frac {\pi }{2}},&{\text{if }}n{\text{ is even}}\\{\frac {n-1}{n}}\cdot {\frac {n-3}{n-2}}\cdots {\frac {4}{5}}\cdot {\frac {2}{3}},&{\text{if }}n{\text{ is odd and more than 1}}\\1,&{\text{if }}n=1\end{cases}}}
सममित सीमाओं के साथ समाकलन
∫ − c c sin x d x = 0 {\displaystyle \int _{-c}^{c}\sin {x}\,dx=0} ∫ − c c cos x d x = 2 ∫ 0 c cos x d x = 2 ∫ − c 0 cos x d x = 2 sin c {\displaystyle \int _{-c}^{c}\cos {x}\,dx=2\int _{0}^{c}\cos {x}\,dx=2\int _{-c}^{0}\cos {x}\,dx=2\sin {c}} ∫ − c c tan x d x = 0 {\displaystyle \int _{-c}^{c}\tan {x}\,dx=0} ∫ − a 2 a 2 x 2 cos 2 n π x a d x = a 3 ( n 2 π 2 − 6 ) 24 n 2 π 2 (for n = 1 , 3 , 5... ) {\displaystyle \int _{-{\frac {a}{2}}}^{\frac {a}{2}}x^{2}\cos ^{2}{\frac {n\pi x}{a}}\,dx={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}}}\qquad {\mbox{(for }}n=1,3,5...{\mbox{)}}} ∫ − a 2 a 2 x 2 sin 2 n π x a d x = a 3 ( n 2 π 2 − 6 ( − 1 ) n ) 24 n 2 π 2 = a 3 24 ( 1 − 6 ( − 1 ) n n 2 π 2 ) (for n = 1 , 2 , 3 , . . . ) {\displaystyle \int _{\frac {-a}{2}}^{\frac {a}{2}}x^{2}\sin ^{2}{\frac {n\pi x}{a}}\,dx={\frac {a^{3}(n^{2}\pi ^{2}-6(-1)^{n})}{24n^{2}\pi ^{2}}}={\frac {a^{3}}{24}}(1-6{\frac {(-1)^{n}}{n^{2}\pi ^{2}}})\qquad {\mbox{(for }}n=1,2,3,...{\mbox{)}}}
एक पूर्ण वृत्त पर अभिन्न
∫ 0 2 π sin 2 m + 1 x cos n x d x = 0 n , m ∈ Z {\displaystyle \int _{0}^{2\pi }\sin ^{2m+1}{x}\cos ^{n}{x}\,dx=0\!\qquad n,m\in \mathbb {Z} } ∫ 0 2 π sin m x cos 2 n + 1 x d x = 0 n , m ∈ Z {\displaystyle \int _{0}^{2\pi }\sin ^{m}{x}\cos ^{2n+1}{x}\,dx=0\!\qquad n,m\in \mathbb {Z} }
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