Triqonometrik funksiyaların inteqralları siyahısı

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Triqonometrik funksiyaların inteqralları siyahısı - bütün Triqonometrik funksiyaların inteqralları haqqında olan düsturları cəmləşdirir. Düsturlardan qeyd etmək lazımdır ki, C (yəni, konstant) heç vaxt sıfra bərabər deyildir.[1]

Əsas Triqonometrik funksiyaların inteqralları[redaktə]

\int \sin(ax+b)\,dx=-\frac1a\cos(ax+b)+C
\int \cos(ax+b)\,dx=\frac1a\sin(ax+b)+C
\int \tan(ax)\,dx=-\frac1a\ln|\cos(ax)|+C=\frac1a\ln| \sec(ax) | + C
\int \operatorname{cotan}(ax)\,dx=\frac1a\ln|\sin(ax)|+C
\int \sin(x)\,dx=-\cos(x)+C
\int \cos(x)\,dx=\sin(x)+C
\int \tan(x)\,dx=-\ln|\cos(x)|+C=\ln| \sec(x) | + C
\int \operatorname{cotan}(x)\,dx=\ln|\sin(x)|+C=-\ln| \operatorname{cosec}(x) | + C

Sinus inteqralları[redaktə]

\int\sin cx\;dx = -\frac{1}{c}\cos cx\,\!
\int\sin^n cx\;dx = -\frac{\sin^{n-1} cx\cos cx}{nc} + \frac{n-1}{n}\int\sin^{n-2} cx\;dx \qquad\mbox{( }n>0\mbox{)}\,\!


\int x\sin cx\;dx = \frac{\sin cx}{c^2}-\frac{x\cos cx}{c}\,\!
\int x^2\sin cx\;dx 
= \frac{2\cos cx}{c^3}
+ \frac{2x\sin cx}{c^2}
- \frac{x^2\cos cx}{c}
\,\!
\int x^3\sin cx\;dx 
=-\frac{6\sin cx}{c^4}
+ \frac{6x\cos cx}{c^3}
+ \frac{3x^2\sin cx}{c^2}
- \frac{x^3\cos cx}{c}
\,\!
\int x^4\sin cx\;dx 
=-\frac{24\cos cx}{c^5}
- \frac{24x\sin cx}{c^4}
+ \frac{12x^2\cos cx}{c^3}
+ \frac{4x^3\sin cx}{c^2}
- \frac{x^4\cos cx}{c}
\,\!
\int x^5\sin cx\;dx 
= \frac{120\sin cx}{c^6}
- \frac{120x\cos cx}{c^5}
- \frac{60x^2\sin cx}{c^4}
+ \frac{20x^3\cos cx}{c^3}
+ \frac{5x^4\sin cx}{c^2}
- \frac{x^5\cos cx}{c}
\,\!

\begin{align}
\int x^n\sin cx\;dx 
& = n! \cdot \sin cx \left[
 \frac{x^{n-1}}{c^2 \cdot (n-1)!}
-\frac{x^{n-3}}{c^4 \cdot (n-3)!}
+\frac{x^{n-5}}{c^6 \cdot (n-5)!} - ...
\right] - \\
& - n! \cdot \cos cx \left[
 \frac{x^n}{c \cdot  n!}
-\frac{x^{n-2}}{c^3 \cdot (n-2)!}
+\frac{x^{n-4}}{c^5 \cdot (n-4)!} - ...
\right]  
\end{align}
\int x^n\sin cx\;dx = -\frac{x^n}{c}\cos cx+\frac{n}{c}\int x^{n-1}\cos cx\;dx \qquad\mbox{( }n>0\mbox{)}\,\!
\int\frac{\sin cx}{x} dx = \sum_{i=0}^\infty (-1)^i\frac{(cx)^{2i+1}}{(2i+1)\cdot (2i+1)!}\,\!
\int\frac{\sin cx}{x^n} dx = -\frac{\sin cx}{(n-1)x^{n-1}} + \frac{c}{n-1}\int\frac{\cos cx}{x^{n-1}} dx\,\!
\int\frac{dx}{\sin cx} = \frac{1}{c}\ln \left|\operatorname{tg}\frac{cx}{2}\right|
\int\frac{dx}{\sin^n cx} = \frac{\cos cx}{c(1-n) \sin^{n-1} cx}+\frac{n-2}{n-1}\int\frac{dx}{\sin^{n-2}cx} \qquad\mbox{( }n>1\mbox{)}\,\!
\int\frac{dx}{1\pm\sin cx} = \frac{1}{c}\operatorname{tg}\left(\frac{cx}{2}\mp\frac{\pi}{4}\right)
\int\frac{x\;dx}{1+\sin cx} = \frac{x}{c}\operatorname{tg}\left(\frac{cx}{2} - \frac{\pi}{4}\right)+\frac{2}{c^2}\ln\left|\cos\left(\frac{cx}{2}-\frac{\pi}{4}\right)\right|
\int\frac{x\;dx}{1-\sin cx} = \frac{x}{c}\operatorname{ctg}\left(\frac{\pi}{4} - \frac{cx}{2}\right)+\frac{2}{c^2}\ln\left|\sin\left(\frac{\pi}{4}-\frac{cx}{2}\right)\right|
\int\frac{\sin cx\;dx}{1\pm\sin cx} = \pm x+\frac{1}{c}\operatorname{tg}\left(\frac{\pi}{4}\mp\frac{cx}{2}\right)
\int\sin c_1x\sin c_2x\;dx = \frac{\sin(c_1-c_2)x}{2(c_1-c_2)}-\frac{\sin(c_1+c_2)x}{2(c_1+c_2)} \qquad\mbox{( }|c_1|\neq|c_2|\mbox{)}\,\!

Kosinus inteqralları[redaktə]

\int\cos cx\;dx = \frac{1}{c}\sin cx\,\!


\int\cos^n cx\;dx = \frac{\cos^{n-1} cx\sin cx}{nc} + \frac{n-1}{n}\int\cos^{n-2} cx\;dx \qquad\mbox{( }n>0\mbox{)}\,\!
\int x\cos cx\;dx = \frac{\cos cx}{c^2} + \frac{x\sin cx}{c}\,\!
\int x^n\cos cx\;dx = \frac{x^n\sin cx}{c} - \frac{n}{c}\int x^{n-1}\sin cx\;dx\,\!
\int\frac{\cos cx}{x} dx = \ln|cx|+\sum_{i=1}^\infty (-1)^i\frac{(cx)^{2i}}{2i\cdot(2i)!}\,\!
\int\frac{\cos cx}{x^n} dx = -\frac{\cos cx}{(n-1)x^{n-1}}-\frac{c}{n-1}\int\frac{\sin cx}{x^{n-1}} dx \qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\frac{dx}{\cos cx} = \frac{1}{c}\ln\left|\operatorname{tg}\left(\frac{cx}{2}+\frac{\pi}{4}\right)\right|
\int\frac{dx}{\cos^n cx} = \frac{\sin cx}{c(n-1) cos^{n-1} cx} + \frac{n-2}{n-1}\int\frac{dx}{\cos^{n-2} cx} \qquad\mbox{( }n>1\mbox{)}\,\!
\int\frac{dx}{1+\cos cx} = \frac{1}{c}\operatorname{tg}\frac{cx}{2}\,\!
\int\frac{dx}{1-\cos cx} = -\frac{1}{c}\operatorname{ctg}\frac{cx}{2}\,\!
\int\frac{x\;dx}{1+\cos cx} = \frac{x}{c}\operatorname{tg}{cx}{2} + \frac{2}{c^2}\ln\left|\cos\frac{cx}{2}\right|
\int\frac{x\;dx}{1-\cos cx} = -\frac{x}{x}\operatorname{ctg}{cx}{2}+\frac{2}{c^2}\ln\left|\sin\frac{cx}{2}\right|
\int\frac{\cos cx\;dx}{1+\cos cx} = x - \frac{1}{c}\operatorname{tg}\frac{cx}{2}\,\!
\int\frac{\cos cx\;dx}{1-\cos cx} = -x-\frac{1}{c}\operatorname{ctg}\frac{cx}{2}\,\!
\int\cos c_1x\cos c_2x\;dx = \frac{\sin(c_1-c_2)x}{2(c_1-c_2)}+\frac{\sin(c_1+c_2)x}{2(c_1+c_2)} \qquad\mbox{( }|c_1|\neq|c_2|\mbox{)}\,\!

Tangens inteqralları[redaktə]

\int\operatorname{tg} cx\;dx = -\frac{1}{c}\ln|\cos cx|\,\!
\int\operatorname{tg}^n cx\;dx = \frac{1}{c(n-1)}\operatorname{tg}^{n-1} cx-\int\operatorname{tg}^{n-2} cx\;dx \qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\frac{dx}{\operatorname{tg} cx + 1} = \frac{x}{2} + \frac{1}{2c}\ln|\sin cx + \cos cx|\,\!
\int\frac{dx}{\operatorname{tg} cx - 1} = -\frac{x}{2} + \frac{1}{2c}\ln|\sin cx - \cos cx|\,\!
\int\frac{\operatorname{tg} cx\;dx}{\operatorname{tg} cx + 1} = \frac{x}{2} - \frac{1}{2c}\ln|\sin cx + \cos cx|\,\!
\int\frac{\operatorname{tg} cx\;dx}{\operatorname{tg} cx - 1} = \frac{x}{2} + \frac{1}{2c}\ln|\sin cx - \cos cx|\,\!

Kotangens inteqralları[redaktə]

\int\operatorname{ctg} cx\;dx = \frac{1}{c}\ln|\sin cx|\,\!
\int\operatorname{ctg}^n cx\;dx = -\frac{1}{c(n-1)}\operatorname{ctg}^{n-1} cx - \int\operatorname{ctg}^{n-2} cx\;dx \qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\frac{dx}{1 + \operatorname{ctg} cx} = \int\frac{\operatorname{tg} cx\;dx}{\operatorname{tg} cx+1}\,\!
\int\frac{dx}{1 - \operatorname{ctg} cx} = \int\frac{\operatorname{tg} cx\;dx}{\operatorname{tg} cx-1}\,\!

Sekans inteqralları[redaktə]

\int \sec{cx} \, dx = \frac{1}{c}\ln{\left| \sec{cx} + \operatorname{tg}{cx}\right|}
\int \sec^n{cx} \, dx = \frac{\sec^{n-1}{cx} \sin {cx}}{c(n-1)} \,+\, \frac{n-2}{n-1}\int \sec^{n-2}{cx} \, dx \qquad \mbox{ ( }n \ne 1\mbox{)}\,\!
\int \frac{dx}{\sec{x} + 1} = x - \operatorname{tg}{\frac{x}{2}}

Kosekans inteqralları[redaktə]

\int \csc{cx} \, dx = -\frac{1}{c}\ln{\left| \csc{cx} + \operatorname{ctg}{cx}\right|}
\int \csc^n{cx} \, dx = -\frac{\csc^{n-1}{cx} \cos{cx}}{c(n-1)} \,+\, \frac{n-2}{n-1}\int \csc^{n-2}{cx} \, dx \qquad \mbox{ ( }n \ne 1\mbox{)}\,\!

Tərkibində yalnız sinus və kosinus olan inteqrallar[redaktə]

\int\frac{dx}{\cos cx\pm\sin cx} = \frac{1}{c\sqrt{2}}\ln\left|\operatorname{tg}\left(\frac{cx}{2}\pm\frac{\pi}{8}\right)\right|
\int\frac{dx}{(\cos cx\pm\sin cx)^2} = \frac{1}{2c}\operatorname{tg}\left(cx\mp\frac{\pi}{4}\right)
\int\frac{dx}{(\cos x + \sin x)^n} = \frac{1}{n-1}\left(\frac{\sin x - \cos x}{(\cos x + \sin x)^{n - 1}} - 2(n - 2)\int\frac{dx}{(\cos x + \sin x)^{n-2}} \right)
\int\frac{\cos cx\;dx}{\cos cx + \sin cx} = \frac{x}{2} + \frac{1}{2c}\ln\left|\sin cx + \cos cx\right|
\int\frac{\cos cx\;dx}{\cos cx - \sin cx} = \frac{x}{2} - \frac{1}{2c}\ln\left|\sin cx - \cos cx\right|
\int\frac{\sin cx\;dx}{\cos cx + \sin cx} = \frac{x}{2} - \frac{1}{2c}\ln\left|\sin cx + \cos cx\right|
\int\frac{\sin cx\;dx}{\cos cx - \sin cx} = -\frac{x}{2} - \frac{1}{2c}\ln\left|\sin cx - \cos cx\right|
\int\frac{\cos cx\;dx}{\sin cx(1+\cos cx)} = -\frac{1}{4c}\operatorname{tg}^2\frac{cx}{2}+\frac{1}{2c}\ln\left|\operatorname{tg}\frac{cx}{2}\right|
\int\frac{\cos cx\;dx}{\sin cx(1+-\cos cx)} = -\frac{1}{4c}\operatorname{ctg}^2\frac{cx}{2}-\frac{1}{2c}\ln\left|\operatorname{tg}\frac{cx}{2}\right|
\int\frac{\sin cx\;dx}{\cos cx(1+\sin cx)} = \frac{1}{4c}\operatorname{ctg}^2\left(\frac{cx}{2}+\frac{\pi}{4}\right)+\frac{1}{2c}\ln\left|\operatorname{tg}\left(\frac{cx}{2}+\frac{\pi}{4}\right)\right|
\int\frac{\sin cx\;dx}{\cos cx(1-\sin cx)} = \frac{1}{4c}\operatorname{tg}^2\left(\frac{cx}{2}+\frac{\pi}{4}\right)-\frac{1}{2c}\ln\left|\operatorname{tg}\left(\frac{cx}{2}+\frac{\pi}{4}\right)\right|
\int\sin cx\cos cx\;dx = \frac{1}{2c}\sin^2 cx\,\!
\int\sin c_1x\cos c_2x\;dx = -\frac{\cos(c_1+c_2)x}{2(c_1+c_2)}-\frac{\cos(c_1-c_2)x}{2(c_1-c_2)} \qquad\mbox{( }|c_1|\neq|c_2|\mbox{)}\,\!
\int\sin^n cx\cos cx\;dx = \frac{1}{c(n+1)}\sin^{n+1} cx \qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\sin cx\cos^n cx\;dx = -\frac{1}{c(n+1)}\cos^{n+1} cx \qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\sin^n cx\cos^m cx\;dx = -\frac{\sin^{n-1} cx\cos^{m+1} cx}{c(n+m)}+\frac{n-1}{n+m}\int\sin^{n-2} cx\cos^m cx\;dx  \qquad\mbox{( }m,n>0\mbox{)}\,\!
\int\sin^n cx\cos^m cx\;dx = \frac{\sin^{n+1} cx\cos^{m-1} cx}{c(n+m)} + \frac{m-1}{n+m}\int\sin^n cx\cos^{m-2} cx\;dx \qquad\mbox{( }m,n>0\mbox{)}\,\!
\int\frac{dx}{\sin cx\cos cx} = \frac{1}{c}\ln\left|\operatorname{tg} cx\right|
\int\frac{dx}{\sin cx\cos^n cx} = \frac{1}{c(n-1)\cos^{n-1} cx}+\int\frac{dx}{\sin cx\cos^{n-2} cx} \qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\frac{dx}{\sin^n cx\cos cx} = -\frac{1}{c(n-1)\sin^{n-1} cx}+\int\frac{dx}{\sin^{n-2} cx\cos cx} \qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\frac{\sin cx\;dx}{\cos^n cx} = \frac{1}{c(n-1)\cos^{n-1} cx} \qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\frac{\sin^2 cx\;dx}{\cos cx} = -\frac{1}{c}\sin cx+\frac{1}{c}\ln\left|\operatorname{tg}\left(\frac{\pi}{4}+\frac{cx}{2}\right)\right|
\int\frac{\sin^2 cx\;dx}{\cos^n cx} = \frac{\sin cx}{c(n-1)\cos^{n-1}cx}-\frac{1}{n-1}\int\frac{dx}{\cos^{n-2}cx} \qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\frac{\sin^n cx\;dx}{\cos cx} = -\frac{\sin^{n-1} cx}{c(n-1)} + \int\frac{\sin^{n-2} cx\;dx}{\cos cx} \qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\frac{\sin^n cx\;dx}{\cos^m cx} = \frac{\sin^{n+1} cx}{c(m-1)\cos^{m-1} cx}-\frac{n-m+2}{m-1}\int\frac{\sin^n cx\;dx}{\cos^{m-2} cx} \qquad\mbox{( }m\neq 1\mbox{)}\,\!
\int\frac{\sin^n cx\;dx}{\cos^m cx} = -\frac{\sin^{n-1} cx}{c(n-m)\cos^{m-1} cx}+\frac{n-1}{n-m}\int\frac{\sin^{n-2} cx\;dx}{\cos^m cx} \qquad\mbox{( }m\neq n\mbox{)}\,\!
\int\frac{\sin^n cx\;dx}{\cos^m cx} = \frac{\sin^{n-1} cx}{c(m-1)\cos^{m-1} cx}-\frac{n-1}{m-1}\int\frac{\sin^{n-1} cx\;dx}{\cos^{m-2} cx} \qquad\mbox{( }m\neq 1\mbox{)}\,\!
\int\frac{\cos cx\;dx}{\sin^n cx} = -\frac{1}{c(n-1)\sin^{n-1} cx} \qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\frac{\cos^2 cx\;dx}{\sin cx} = \frac{1}{c}\left(\cos cx+\ln\left|\operatorname{tg}\frac{cx}{2}\right|\right)
\int\frac{\cos^2 cx\;dx}{\sin^n cx} = -\frac{1}{n-1}\left(\frac{\cos cx}{c\sin^{n-1} cx)}+\int\frac{dx}{\sin^{n-2} cx}\right) \qquad\mbox{( }n\neq 1\mbox{)}
\int\frac{\cos^n cx\;dx}{\sin^m cx} = -\frac{\cos^{n+1} cx}{c(m-1)\sin^{m-1} cx} - \frac{n-m-2}{m-1}\int\frac{cos^n cx\;dx}{\sin^{m-2} cx} \qquad\mbox{( }m\neq 1\mbox{)}\,\!
\int\frac{\cos^n cx\;dx}{\sin^m cx} = \frac{\cos^{n-1} cx}{c(n-m)\sin^{m-1} cx} + \frac{n-1}{n-m}\int\frac{cos^{n-2} cx\;dx}{\sin^m cx} \qquad\mbox{( }m\neq n\mbox{)}\,\!
\int\frac{\cos^n cx\;dx}{\sin^m cx} = -\frac{\cos^{n-1} cx}{c(m-1)\sin^{m-1} cx} - \frac{n-1}{m-1}\int\frac{cos^{n-2} cx\;dx}{\sin^{m-2} cx} \qquad\mbox{( }m\neq 1\mbox{)}\,\!

Tərkibində yalnız sinus və tangens olan inteqrallar[redaktə]

\int \sin cx \operatorname{tg} cx\;dx = \frac{1}{c}(\ln|\sec cx + \operatorname{tg} cx| - \sin cx)\,\!
\int\frac{\operatorname{tg}^n cx\;dx}{\sin^2 cx} = \frac{1}{c(n-1)}\operatorname{tg}^{n-1} (cx) \qquad\mbox{( }n\neq 1\mbox{)}\,\!

Tərkibində yalnız kosinus və tangens olan inteqrallar[redaktə]

\int\frac{\operatorname{tg}^n cx\;dx}{\cos^2 cx} = \frac{1}{c(n+1)}\operatorname{tg}^{n+1} cx \qquad\mbox{( }n\neq -1\mbox{)}\,\!

Tərkibində yalnız sinus və kotangens olan inteqrallar[redaktə]

\int\frac{\operatorname{ctg}^n cx\;dx}{\sin^2 cx} = \frac{1}{c(n+1)}\operatorname{ctg}^{n+1} cx  \qquad\mbox{( }n\neq -1\mbox{)}\,\!

Tərkibində yalnız kosinus və kotangens olan inteqrallar[redaktə]

\int\frac{\operatorname{ctg}^n cx\;dx}{\cos^2 cx} = \frac{1}{c(1-n)}\operatorname{tg}^{1-n} cx \qquad\mbox{( }n\neq 1\mbox{)}\,\!

Tərkibində yalnız tangens və kotangens olan inteqrallar[redaktə]

\int \frac{\operatorname{tg}^m(cx)}{\operatorname{ctg}^n(cx)}\;dx = \frac{1}{c(m+n-1)}\operatorname{tg}^{m+n-1}(cx) - \int \frac{\operatorname{tg}^{m-2}(cx)}{\operatorname{ctg}^n(cx)}\;dx\qquad\mbox{( }m + n \neq 1\mbox{)}\,\!

Simmetrik limitlərin inteqralları[redaktə]

\int_{{-c}}^{{c}}\sin {x}\;dx = 0 \!
\int_{{-c}}^{{c}}\cos {x}\;dx = 2\int_{{0}}^{{c}}\cos {x}\;dx = 2\int_{{-c}}^{{0}}\cos {x}\;dx = 2\sin {c} \!
\int_{{-c}}^{{c}}\tan {x}\;dx = 0 \!
\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{)}\,\!

Simmetrik funksiyaların inteqralları[redaktə]

\int_{{-c}}^{{c}}\sin {x}\;dx = 0 \!
\int_{{-c}}^{{c}}\cos {x}\;dx = 2\int_{{0}}^{{c}}\cos {x}\;dx = 2\int_{{-c}}^{{0}}\cos {x}\;dx \!
\int_{{-c}}^{{c}}\tan {x}\;dx = 0 \!
\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{(voor }n=1,3,5...\mbox{)}\,\!

İstinadlar[redaktə]

  1. Stewart, James. Calculus: Early Transcendentals, 6th Edition. Thomson: 2008