Całki funkcji trygonometrycznych

Całki funkcji tryg. zawierających sinus

$\int \sin cxdx=-\frac{1}{c}\cos cx$

$\int {\sin }^{n}cxdx=-\frac{{\sin }^{n-1}cx\cos cx}{nc}+\frac{n-1}{n}\int {\sin }^{n-2}cxdx\text{(dla }n>0\text{)}$

$\int x\sin cxdx=\frac{\sin cx}{c^2}-\frac{x\cos cx}{c}$

$\int x^n\sin cxdx=-\frac{x^n}{c}\cos cx+\frac{n}{c}\int x^{n-1}\cos cxdx\text{(dla }n>0\text{)}$

$\int \frac{\sin cx}{x}dx=\sum _{i=0}^{\mathrm{\infty }}(-1{)}^i\frac{(cx{)}^{2i+1}}{(2i+1)\cdot (2i+1)!}$

$\int \frac{\sin cx}{x^n}dx=-\frac{\sin cx}{(n-1)x^{n-1}}+\frac{c}{n-1}\int \frac{\cos cx}{x^{n-1}}dx$

$\int \frac{dx}{\sin cx}=\frac{1}{c}\ln |\mathrm{tg}\frac{cx}{2}|$

$\int \frac{dx}{{\sin }^{n}cx}=\frac{\cos cx}{c(1-n){\sin }^{n-1}cx}+\frac{n-2}{n-1}\int \frac{dx}{{\sin }^{n-2}cx}\text{(dla }n>1\text{)}$

$\int \frac{dx}{1\pm \sin cx}=\frac{1}{c}\mathrm{tg}(\frac{cx}{2}\mp \frac{\pi }{4})$

$\int \frac{xdx}{1+\sin cx}=\frac{x}{c}\mathrm{tg}(\frac{cx}{2}-\frac{\pi }{4})+\frac{2}{c^2}\ln |\cos (\frac{cx}{2}-\frac{\pi }{4})|$

$\int \frac{xdx}{1-\sin cx}=\frac{x}{c}\mathrm{ctg}(\frac{\pi }{4}-\frac{cx}{2})+\frac{2}{c^2}\ln |\sin (\frac{\pi }{4}-\frac{cx}{2})|$

$\int \frac{\sin cxdx}{1\pm \sin cx}=\pm x+\frac{1}{c}\mathrm{tg}(\frac{\pi }{4}\mp \frac{cx}{2})$

$\int \sin c_{1}x\sin c_{2}xdx=\frac{\sin (c_1-c_2)x}{2(c_1-c_2)}-\frac{\sin (c_1+c_2)x}{2(c_1+c_2)}\text{(dla }|c_1|\ne |c_2|\text{)}$

Całki funkcji tryg. zawierających cosinus

$\int \cos cxdx=\frac{1}{c}\sin cx$

$\int {\cos }^{n}cxdx=\frac{{\cos }^{n-1}cx\sin cx}{nc}+\frac{n-1}{n}\int {\cos }^{n-2}cxdx\text{(dla }n>0\text{)}$

$\int x\cos cxdx=\frac{\cos cx}{c^2}+\frac{x\sin cx}{c}$

$\int x^n\cos cxdx=\frac{x^n\sin cx}{c}-\frac{n}{c}\int x^{n-1}\sin cxdx$

$\int \frac{\cos cx}{x}dx=\ln |cx|+\sum _{i=1}^{\mathrm{\infty }}(-1{)}^i\frac{(cx{)}^{2i}}{2i\cdot (2i)!}$

$\int \frac{\cos cx}{x^n}dx=-\frac{\cos cx}{(n-1)x^{n-1}}-\frac{c}{n-1}\int \frac{\sin cx}{x^{n-1}}dx\text{(dla }n\ne 1\text{)}$

$\int \frac{dx}{\cos cx}=\frac{1}{c}\ln |\mathrm{tg}(\frac{cx}{2}+\frac{\pi }{4})|$

$\int \frac{dx}{{\cos }^{n}cx}=\frac{\sin cx}{c(n-1)co{s}^{n-1}cx}+\frac{n-2}{n-1}\int \frac{dx}{{\cos }^{n-2}cx}\text{(dla }n>1\text{)}$

$\int \frac{dx}{1+\cos cx}=\frac{1}{c}\mathrm{tg}\frac{cx}{2}$

$\int \frac{dx}{1-\cos cx}=-\frac{1}{c}\mathrm{ctg}\frac{cx}{2}$

$\int \frac{xdx}{1+\cos cx}=\frac{x}{c}\mathrm{tg}\frac{cx}{2}+\frac{2}{c^2}\ln |\cos \frac{cx}{2}|$

$\int \frac{xdx}{1-\cos cx}=-\frac{x}{c}\mathrm{ctg}\frac{cx}{2}+\frac{2}{c^2}\ln |\sin \frac{cx}{2}|$

$\int \frac{\cos cxdx}{1+\cos cx}=x-\frac{1}{c}\mathrm{tg}\frac{cx}{2}$

$\int \frac{\cos cxdx}{1-\cos cx}=-x-\frac{1}{c}\mathrm{ctg}\frac{cx}{2}$

$\int \cos c_{1}x\cos c_{2}xdx=\frac{\sin (c_1-c_2)x}{2(c_1-c_2)}+\frac{\sin (c_1+c_2)x}{2(c_1+c_2)}\text{(dla }|c_1|\ne |c_2|\text{)}$

Całki funkcji tryg. zawierających tangens

$\int \mathrm{tg}cxdx=-\frac{1}{c}\ln |\cos cx|$

$\int {\mathrm{tg}}^{n}cxdx=\frac{1}{c(n-1)}{\mathrm{tg}}^{n-1}cx-\int {\mathrm{tg}}^{n-2}cxdx\text{(dla }n\ne 1\text{)}$

$\int \frac{dx}{\mathrm{tg}cx+1}=\frac{x}{2}+\frac{1}{2c}\ln |\sin cx+\cos cx|$

$\int \frac{dx}{\mathrm{tg}cx-1}=-\frac{x}{2}+\frac{1}{2c}\ln |\sin cx-\cos cx|$

$\int \frac{\mathrm{tg}cxdx}{\mathrm{tg}cx+1}=\frac{x}{2}-\frac{1}{2c}\ln |\sin cx+\cos cx|$

$\int \frac{\mathrm{tg}cxdx}{\mathrm{tg}cx-1}=\frac{x}{2}+\frac{1}{2c}\ln |\sin cx-\cos cx|$

Całki funkcji tryg. zawierających secans

$\int \sec cxdx=\frac{1}{c}\ln |\sec cx+\mathrm{tg}cx|$

$\int {\sec }^{n}cxdx=\frac{{\sec }^{n-1}cx\sin cx}{c(n-1)}+\frac{n-2}{n-1}\int {\sec }^{n-2}cxdx\text{ dla }n\ne 1,c\ne 0$

Całki funkcji tryg. zawierających cosecans

$\int \csc cxdx=-\frac{1}{c}\ln |\csc cx+\mathrm{ctg}cx|$

$\int {\csc }^{n}cxdx=-\frac{{\csc }^{n-1}cx\cos cx}{c(n-1)}+\frac{n-2}{n-1}\int {\csc }^{n-2}cxdx\text{ dla }n\ne 1,c\ne 0$

Całki funkcji tryg. zawierających cotangens

$\int \mathrm{ctg}cxdx=\frac{1}{c}\ln |\sin cx|$

$\int {\mathrm{ctg}}^{n}cxdx=-\frac{1}{c(n-1)}{\mathrm{ctg}}^{n-1}cx-\int {\mathrm{ctg}}^{n-2}cxdx\text{(dla }n\ne 1\text{)}$

$\int \frac{dx}{1+\mathrm{ctg}cx}=\int \frac{\mathrm{tg}cxdx}{\mathrm{tg}cx+1}$

$\int \frac{dx}{1-\mathrm{ctg}cx}=\int \frac{\mathrm{tg}cxdx}{\mathrm{tg}cx-1}$

Całki funkcji tryg. zawierających sinus i cosinus

$\int \frac{dx}{\cos cx\pm \sin cx}=\frac{1}{c\sqrt{2}}\ln |\mathrm{tg}(\frac{cx}{2}\pm \frac{\pi }{8})|$

$\int \frac{dx}{(\cos cx\pm \sin cx{)}^2}=\frac{1}{2c}\mathrm{tg}(cx\mp \frac{\pi }{4})$

$\int \frac{\cos cxdx}{\cos cx+\sin cx}=\frac{x}{2}+\frac{1}{2c}\ln |\sin cx+\cos cx|$

$\int \frac{\cos cxdx}{\cos cx-\sin cx}=\frac{x}{2}-\frac{1}{2c}\ln |\sin cx-\cos cx|$

$\int \frac{\sin cxdx}{\cos cx+\sin cx}=\frac{x}{2}-\frac{1}{2c}\ln |\sin cx+\cos cx|$

$\int \frac{\sin cxdx}{\cos cx-\sin cx}=-\frac{x}{2}-\frac{1}{2c}\ln |\sin cx-\cos cx|$

$\int \frac{\cos cxdx}{\sin cx(1+\cos cx)}=-\frac{1}{4c}{\mathrm{tg}}^2\frac{cx}{2}+\frac{1}{2c}\ln |\mathrm{tg}\frac{cx}{2}|$

$\int \frac{\cos cxdx}{\sin cx(1+-\cos cx)}=-\frac{1}{4c}{\mathrm{ctg}}^2\frac{cx}{2}-\frac{1}{2c}\ln |\mathrm{tg}\frac{cx}{2}|$

$\int \frac{\sin cxdx}{\cos cx(1+\sin cx)}=\frac{1}{4c}{\mathrm{ctg}}^2(\frac{cx}{2}+\frac{\pi }{4})+\frac{1}{2c}\ln |\mathrm{tg}(\frac{cx}{2}+\frac{\pi }{4})|$

$\int \frac{\sin cxdx}{\cos cx(1-\sin cx)}=\frac{1}{4c}{\mathrm{tg}}^2(\frac{cx}{2}+\frac{\pi }{4})-\frac{1}{2c}\ln |\mathrm{tg}(\frac{cx}{2}+\frac{\pi }{4})|$

$\int \sin cx\cos cxdx=\frac{1}{2c}{\sin }^{2}cx$

$\int \sin c_{1}x\cos c_{2}xdx=-\frac{\cos (c_1+c_2)x}{2(c_1+c_2)}-\frac{\cos (c_1-c_2)x}{2(c_1-c_2)}\text{(dla }|c_1|\ne |c_2|\text{)}$

$\int {\sin }^{n}cx\cos cxdx=\frac{1}{c(n+1)}{\sin }^{n+1}cx\text{(dla }n\ne 1\text{)}$

$\int \sin cx{\cos }^{n}cxdx=-\frac{1}{c(n+1)}{\cos }^{n+1}cx\text{(dla }n\ne 1\text{)}$

$\int {\sin }^{n}cx{\cos }^{m}cxdx=-\frac{{\sin }^{n-1}cx{\cos }^{m+1}cx}{c(n+m)}+\frac{n-1}{n+m}\int {\sin }^{n-2}cx{\cos }^{m}cxdx\text{(dla }m,n>0\text{)}$

również: $\int {\sin }^{n}cx{\cos }^{m}cxdx=\frac{{\sin }^{n+1}cx{\cos }^{m-1}cx}{c(n+m)}+\frac{m-1}{n+m}\int {\sin }^{n}cx{\cos }^{m-2}cxdx\text{(dla }m,n>0\text{)}$

$\int \frac{dx}{\sin cx\cos cx}=\frac{1}{c}\ln |\mathrm{tg}cx|$

$\int \frac{dx}{\sin cx{\cos }^{n}cx}=\frac{1}{c(n-1){\cos }^{n-1}cx}+\int \frac{dx}{\sin cx{\cos }^{n-2}cx}\text{(dla }n\ne 1\text{)}$

$\int \frac{dx}{{\sin }^{n}cx\cos cx}=-\frac{1}{c(n-1){\sin }^{n-1}cx}+\int \frac{dx}{{\sin }^{n-2}cx\cos cx}\text{(dla }n\ne 1\text{)}$

$\int \frac{\sin cxdx}{{\cos }^{n}cx}=\frac{1}{c(n-1){\cos }^{n-1}cx}\text{(dla }n\ne 1\text{)}$

$\int \frac{{\sin }^{2}cxdx}{\cos cx}=-\frac{1}{c}\sin cx+\frac{1}{c}\ln |\mathrm{tg}(\frac{\pi }{4}+\frac{cx}{2})|$

$\int \frac{{\sin }^{2}cxdx}{{\cos }^{n}cx}=\frac{\sin cx}{c(n-1){\cos }^{n-1}cx}-\frac{1}{n-1}\int \frac{dx}{{\cos }^{n-2}cx}\text{(dla }n\ne 1\text{)}$

$\int \frac{{\sin }^{n}cxdx}{\cos cx}=-\frac{{\sin }^{n-1}cx}{c(n-1)}+\int \frac{{\sin }^{n-2}cxdx}{\cos cx}\text{(dla }n\ne 1\text{)}$

$\int \frac{si{n}^{n}cxdx}{{\cos }^{m}cx}=\frac{{\sin }^{n+1}cx}{c(m-1){\cos }^{m-1}cx}-\frac{n-m+2}{m-1}\int \frac{{\sin }^{n}cxdx}{{\cos }^{m-2}cx}\text{(dla }m\ne 1\text{)}$

również: $\int \frac{si{n}^{n}cxdx}{{\cos }^{m}cx}=-\frac{{\sin }^{n-1}cx}{c(n-m){\cos }^{m-1}cx}+\frac{n-1}{n-m}\int \frac{{\sin }^{n-2}cxdx}{{\cos }^{m}cx}\text{(dla }m\ne n\text{)}$

również: $\int \frac{si{n}^{n}cxdx}{{\cos }^{m}cx}=\frac{{\sin }^{n-1}cx}{c(m-1){\cos }^{m-1}cx}-\frac{n-1}{n-1}\int \frac{{\sin }^{n-1}cxdx}{{\cos }^{m-2}cx}\text{(dla }m\ne 1\text{)}$

$\int \frac{\cos cxdx}{{\sin }^{n}cx}=-\frac{1}{c(n-1){\sin }^{n-1}cx}\text{(dla }n\ne 1\text{)}$

$\int \frac{{\cos }^{2}cxdx}{\sin cx}=\frac{1}{c}(\cos cx+\ln |\mathrm{tg}\frac{cx}{2}|)$

$\int \frac{{\cos }^{2}cxdx}{{\sin }^{n}cx}=-\frac{1}{n-1}(\frac{\cos cx}{c{\sin }^{n-1}cx)}+\int \frac{dx}{{\sin }^{n-2}cx})\text{(dla }n\ne 1\text{)}$

$\int \frac{{\cos }^{n}cxdx}{{\sin }^{m}cx}=-\frac{{\cos }^{n+1}cx}{c(m-1){\sin }^{m-1}cx}-\frac{n-m-2}{m-1}\int \frac{co{s}^{n}cxdx}{{\sin }^{m-2}cx}\text{(dla }m\ne 1\text{)}$

również: $\int \frac{{\cos }^{n}cxdx}{{\sin }^{m}cx}=\frac{{\cos }^{n-1}cx}{c(n-m){\sin }^{m-1}cx}+\frac{n-1}{n-m}\int \frac{co{s}^{n-2}cxdx}{{\sin }^{m}cx}\text{(dla }m\ne n\text{)}$

również: $\int \frac{{\cos }^{n}cxdx}{{\sin }^{m}cx}=-\frac{{\cos }^{n-1}cx}{c(m-1){\sin }^{m-1}cx}-\frac{n-1}{m-1}\int \frac{co{s}^{n-2}cxdx}{{\sin }^{m-2}cx}\text{(dla }m\ne 1\text{)}$

Całki funkcji tryg. zawierających sinus i tangens

$\int \sin cx\mathrm{tg}cxdx=\frac{1}{c}(\ln |\sec cx+\mathrm{tg}cx|-\sin cx)$

$\int \frac{{\mathrm{tg}}^{n}cxdx}{{\sin }^{2}cx}=\frac{1}{c(n-1)}{\mathrm{tg}}^{n-1}(cx)\text{(dla }n\ne 1\text{)}$

Całki funkcji tryg. zawierających cosinus i tangens

$\int \frac{{\mathrm{tg}}^{n}cxdx}{{\cos }^{2}cx}=\frac{1}{c(n+1)}{\mathrm{tg}}^{n+1}cx\text{(dla }n\ne -1\text{)}$

Całki funkcji tryg. zawierających sinus i cotangens

$\int \frac{{\mathrm{ctg}}^{n}cxdx}{{\sin }^{2}cx}=\frac{1}{c(n+1)}{\mathrm{ctg}}^{n+1}cx\text{(dla }n\ne -1\text{)}$

Całki funkcji tryg. zawierających cosinus i cotangens

$\int \frac{{\mathrm{ctg}}^{n}cxdx}{{\cos }^{2}cx}=\frac{1}{c(1-n)}{\mathrm{tg}}^{1-n}cx\text{(dla }n\ne 1\text{)}$

Całki funkcji tryg. zawierających tangens i cotangens

$\int \frac{{\mathrm{tg}}^m(cx)}{{\mathrm{ctg}}^n(cx)}dx=\frac{1}{c(m+n-1)}{\mathrm{tg}}^{m+n-1}(cx)-\int \frac{{\mathrm{tg}}^{m-2}(cx)}{{\mathrm{ctg}}^n(cx)}dx\text{(dla }m+n\ne 1\text{)}$

Szablon:PD-ineligible-tekst