Тригонометрилле функцисен интегралĕсен йышĕ

Тригонометрилле функцисен интегралĕсен йышĕ. Аяларах çавнашкал интегралсен (умсăнарсен) йышне илсе кăтартнă. Пур çĕрте те аддитивлă констаттăна катертнĕ.

Ĕнтĕ тата $c$ кнстанттă нуль мар.

Ĕнтĕ синус çеç пур интегралсем

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

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

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

$\int x^2\sin cxdx=\frac{2\cos cx}{c^3}+\frac{2x\sin cx}{c^2}-\frac{x^2\cos cx}{c}$

$\int x^3\sin cxdx=-\frac{6\sin cx}{c^4}+\frac{6x\cos cx}{c^3}+\frac{3x^2\sin cx}{c^2}-\frac{x^3\cos cx}{c}$

$\int x^4\sin cxdx=-\frac{24\cos cx}{c^5}-\frac{24x\sin cx}{c^4}+\frac{12x^2\cos cx}{c^3}+\frac{4x^3\sin cx}{c^2}-\frac{x^4\cos cx}{c}$

$\int x^5\sin cxdx=\frac{120\sin cx}{c^6}-\frac{120x\cos cx}{c^5}-\frac{60x^2\sin cx}{c^4}+\frac{20x^3\cos cx}{c^3}+\frac{5x^4\sin cx}{c^2}-\frac{x^5\cos cx}{c}$

$\begin{array}{cc}\int x^n\sin cxdx& =n!\cdot \sin cx[\frac{x^{n-1}}{c^2\cdot (n-1)!}-\frac{x^{n-3}}{c^4\cdot (n-3)!}+\frac{x^{n-5}}{c^6\cdot (n-5)!}-...]-\\ & -n!\cdot \cos cx[\frac{x^n}{c\cdot n!}-\frac{x^{n-2}}{c^3\cdot (n-2)!}+\frac{x^{n-4}}{c^5\cdot (n-4)!}-...]\end{array}$

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

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

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

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

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

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

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

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

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

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

Ĕнтĕ косинус çеç пур интегралсем

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Ĕнтĕ тангенс çеç пур интегралсем

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

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

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

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

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

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

Ĕнтĕ секанс çеç пур интегралсем

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

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

$\int \frac{dx}{\sec x+1}=x-\mathrm{tg}\frac{x}{2}$

Ĕнтĕ косеканс çеç пур интегралсем

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

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

Ĕнтĕ котангенс çеç пур интегралсем

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

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

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

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

Ĕнтĕ синус тата косинус çеç пур интегралсем

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

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

$\int \frac{dx}{(\cos x+\sin x{)}^n}=\frac{1}{n-1}(\frac{\sin x-\cos x}{(\cos x+\sin x{)}^{n-1}}-2(n-2)\int \frac{dx}{(\cos x+\sin x{)}^{n-2}})$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Ĕнтĕ синус тата тангенс çеç пур интегралсем

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

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

Ĕнтĕ косинус тата тангенс çеç пур интегралсем

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

Ĕнтĕ синус тата котангенс çеç пур интегралсем

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

Ĕнтĕ косинус тата котангенс çеç пур интегралсем

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

Ĕнтĕ тангенс тата котангенс çеç пур интегралсем

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

Библиографи

Кĕнекесем

• Градштейн И. С. Рыжик И. М. Таблицы интегралов, сумм, рядов и произведений. — 4-е изд. — М.: Наука, 1963. — ISBN 0-12-294757-6 // EqWorld
• Двайт Г. Б. Таблицы интегралов СПб: Издательство и типография АО ВНИИГ им. Б. В. Веденеева, 1995. — 176 с. — ISBN 5-85529-029-8.
• D. Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st ed., 2002. ISBN 1-58488-291-3.
• M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1964. ISBN 0-486-61272-4

Интегралсен таблицисем

• Интегралы на EqWorld
• S.O.S. Mathematics: Tables and Formulas

Интегралсене шутлани

• The Integrator (Wolfram Research)
• Империя Чисел
• Методы вычисления неопределённых интегралов