त्रिकोणमितीय फलनों के अभिन्नों की सूची
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{{Trigonometry}}
{{Trigonometry}}
'''[[त्रिकोणमितीय कार्य|त्रिकोणमितीय फलनों]] [[अभिन्न|समाकलन]] ([[ antiderivative |प्रतिअवकलन]] [[फ़ंक्शन (गणित)|फलन (गणित)]]) की सूची''' निम्नलिखित है। घातांकीय और त्रिकोणमितीय दोनों फलनों से जुड़े प्रतिअवकलन के लिए, घातांकीय फलनों के [[अभिन्नों की सूची|समाकलनों की सूची]] देखें। प्रतिअवकलन फलनों की पूर्ण सूची के लिए, समाकलनों की सूचियाँ देखें। त्रिकोणमितीय फलनों से जुड़े विशेष प्रतिअवकलन के लिए, [[त्रिकोणमितीय अभिन्न|त्रिकोणमितीय समाकलन]] भाग देखें।
'''[[त्रिकोणमितीय कार्य|त्रिकोणमितीय फलनों]] [[अभिन्न|समाकलन]] ([[ antiderivative |प्रतिअवकलन]] [[फ़ंक्शन (गणित)|फलन (गणित)]]) की सूची''' निम्नलिखित है। घातांकीय और त्रिकोणमितीय दोनों फलनों से जुड़े प्रतिअवकलन के लिए, घातांकीय फलनों के [[अभिन्नों की सूची|समाकलनों की सूची]] देखें। प्रतिअवकलन फलनों की पूर्ण सूची के लिए, समाकलनों की सूचियाँ देखें। त्रिकोणमितीय फलनों से जुड़े विशेष प्रतिअवकलन के लिए, [[त्रिकोणमितीय अभिन्न|त्रिकोणमितीय समाकलन]] भाग देखें।
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Revision as of 13:41, 2 August 2023
त्रिकोणमितीय फलनों समाकलन (प्रतिअवकलन फलन (गणित) ) की सूचीसमाकलनों की सूची देखें। प्रतिअवकलन फलनों की पूर्ण सूची के लिए, समाकलनों की सूचियाँ देखें। त्रिकोणमितीय फलनों से जुड़े विशेष प्रतिअवकलन के लिए, त्रिकोणमितीय समाकलन भाग देखें।
सामान्यतः, यदि फलन 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 साइन सम्मिलित 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 सम्मिलित 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) सम्मिलित है सेकेंट फलन 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 सम्मिलित 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 सम्मिलित 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 सम्मिलित करने वाला समाकलन समाकलन भाग जो साइन और कोसाइन का तर्कसंगत फलन 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 सम्मिलित 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 सम्मिलित 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 सम्मिलित 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 सम्मिलित 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 सम्मिलित 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 सम्मिलित 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 फलन 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{)}}}
पूर्ण वृत्त पर समाकलन
∫ 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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श्रेणी:समाकलनों की सूचियाँ
श्रेणी:त्रिकोणमिति
