فرمول های انتگرال (Integral) شامل توابع مثلثاتی (Trigonometric Function)، در ریاضیات (Mathematics)

سری اول : *

\[ \int {\sin } axdx = - {1 \over a}\cos ax + C \] \[ \int {\cos } axdx = {1 \over a}\sin ax + C \] \[ \int {{{\sin }^2}} axdx = {x \over 2} - {{\sin 2ax} \over {4a}} + C \] \[ \int {{{\cos }^2}} axdx = {x \over 2} + {{\sin 2ax} \over {4a}} + C \] \[ \int {{{\sin }^n}} axdx = - {{{{\sin }^{n - 1}}ax\cos ax} \over {na}} + {{n - 1} \over n}\int {{{\sin }^{n - 2}}} axdx \] \[ \int {{{\cos }^n}} axdx = {{{{\cos }^{n - 1}}ax\sin ax} \over {na}} + {{n - 1} \over n}\int {{{\cos }^{n - 2}}} axdx \] \[ \int {\sin } ax\cos bxdx = - {{\cos (a + b)x} \over {2(a + b)}} - {{\cos (a - b)x} \over {2(a - b)}} + C \ \ \ \ ,{a^2} \ne {b^2} \] \[ \int {\sin } ax\sin bxdx = {{\sin (a - b)x} \over {2(a - b)}} - {{\sin (a + b)x} \over {2(a + b)}} + C \ \ \ \ ,{a^2} \ne {b^2} \] \[ \int {\cos } ax\cos bxdx = {{\sin (a - b)x} \over {2(a - b)}} + {{\sin (a + b)x} \over {2(a + b)}} + C \ \ \ \ ,{a^2} \ne {b^2} \] \[ \int {\sin } ax\cos axdx = - {{\cos 2ax} \over {4a}} + C \] \[ \int {{{\sin }^n}} ax\cos axdx = {{{{\sin }^{n + 1}}ax} \over {(n + 1)a}} + C \ \ \ \ ,n \ne - 1 \] \[ \int {{{\cos ax} \over {\sin ax}}} dx = {1 \over a}\ln |\sin ax| + C \] \[ \int {{{\cos }^n}} ax\sin axdx = - {{{{\cos }^{n + 1}}ax} \over {(n + 1)a}} + C \ \ \ \ ,n \ne - 1 \] \[ \int {{{\sin ax} \over {\cos ax}}} dx = - {1 \over a}\ln |\cos ax| + C \] \[ \displaylines{ \int {{{\sin }^n}} ax{\cos ^m}axdx = - {{{{\sin }^{n - 1}}ax{{\cos }^{m + 1}}ax} \over {a(m + n)}} + {{n - 1} \over {m + n}}\int {{{\sin }^{n - 2}}} ax{\cos ^m}axdx \cr ,n \ne - m \ \ \ \ (reduces \ \ {\sin ^n}ax) \cr} \] \[ \displaylines{ \int {{{\sin }^n}} ax{\cos ^m}axdx = {{{{\sin }^{n + 1}}ax{{\cos }^{m - 1}}ax} \over {a(m + n)}} + {{m - 1} \over {m + n}}\int {{{\sin }^n}} ax{\cos ^{m - 2}}axdx \cr ,m \ne - n \ \ \ \ (reduces \ \ {\cos ^m}ax) \cr} \] \[ \int {{{dx} \over {b + c\sin ax}}} = {{ - 2} \over {a\sqrt {{b^2} - {c^2}} }}{\tan ^{ - 1}}\left[ {\sqrt {{{b - c} \over {b + c}}} \tan ({\pi \over 4} - {{ax} \over 2})} \right] + C \ \ \ \ ,{b^2} > {c^2} \] \[ \int {{{dx} \over {b + c\sin ax}}} = {{ - 1} \over {a\sqrt {{c^2} - {b^2}} }}\ln \left| {{{c + b\sin ax + \sqrt {{c^2} - {b^2}} \cos ax} \over {b + c\sin ax}}} \right| + C \ \ \ \ ,{b^2} < {c^2} \] \[ \int {{{dx} \over {1 + \sin ax}}} = - {1 \over a}\tan ({\pi \over 4} - {{ax} \over 2}) + C \] \[ \int {{{dx} \over {1 - \sin ax}}} = {1 \over a}\tan ({\pi \over 4} + {{ax} \over 2}) + C \] \[ \int {{{dx} \over {b + c\cos ax}}} = {2 \over {a\sqrt {{b^2} - {c^2}} }}{\tan ^{ - 1}}\left[ {\sqrt {{{b - c} \over {b + c}}} \tan {{ax} \over 2}} \right] + C \ \ \ \ ,{b^2} > {c^2} \] \[ \int {{{dx} \over {b + c\cos ax}}} = {1 \over {a\sqrt {{c^2} - {b^2}} }}\ln \left| {{{c + b\cos ax + \sqrt {{c^2} - {b^2}} \sin ax} \over {b + c\cos ax}}} \right| + C \ \ \ \ ,{b^2} < {c^2} \] \[ \int {{{dx} \over {1 + \cos ax}}} = {1 \over a}\tan {{ax} \over 2} + C \] \[ \int {{{dx} \over {1 - \cos ax}}} = - {1 \over a}\cot {{ax} \over 2} + C \] \[ \int x \sin axdx = {1 \over {{a^2}}}\sin ax - {x \over a}\cos ax + C \] \[ \int x \cos axdx = {1 \over {{a^2}}}\cos ax + {x \over a}\sin ax + C \] \[ \int {{x^n}} \sin axdx = - {{{x^n}} \over a}\cos ax + {n \over a}\int {{x^{n - 1}}} \cos axdx \] \[ \int {{x^n}} \cos axdx = {{{x^n}} \over a}\sin ax - {n \over a}\int {{x^{n - 1}}} \sin axdx \] \[ \int {\tan } axdx = {1 \over a}\ln |\sec ax| + C \] \[ \int {\cot } axdx = {1 \over a}\ln |\sin ax| + C \] \[ \int {{{\tan }^2}} axdx = {1 \over a}\tan ax - x + C \] \[ \int {{{\cot }^2}} axdx = - {1 \over a}\cot ax - x + C \] \[ \int {{{\tan }^n}} axdx = {{{{\tan }^{n - 1}}ax} \over {a(n - 1)}} - \int {{{\tan }^{n - 2}}} axdx \ \ \ \ ,n \ne 1 \] \[ \int {{{\cot }^n}} axdx = - {{{{\cot }^{n - 1}}ax} \over {a(n - 1)}} - \int {{{\cot }^{n - 2}}} axdx \ \ \ \ ,n \ne 1 \] \[ \int {\sec } axdx = {1 \over a}\ln |\sec ax + \tan ax| + C \] \[ \int {\csc } axdx = - {1 \over a}\ln |\csc ax + \cot ax| + C \] \[ \int {{{\sec }^2}} axdx = {1 \over a}\tan ax + C \] \[ \int {{{\csc }^2}} axdx = - {1 \over a}\cot ax + C \] \[ \int {{{\sec }^n}} axdx = {{{{\sec }^{n - 2}}ax\tan ax} \over {a(n - 1)}} + {{n - 2} \over {n - 1}}\int {{{\sec }^{n - 2}}} axdx \ \ \ \ ,n \ne 1 \] \[ \int {{{\csc }^n}} axdx = - {{{{\csc }^{n - 2}}ax\cot ax} \over {a(n - 1)}} + {{n - 2} \over {n - 1}}\int {{{\csc }^{n - 2}}} axdx \ \ \ \ ,n \ne 1 \] \[ \int {{{\sec }^n}} ax\tan axdx = {{{{\sec }^n}ax} \over {na}} + C \ \ \ \ ,n \ne 0 \] \[ \int {{{\csc }^n}} ax\cot axdx = - {{{{\csc }^n}ax} \over {na}} + C \ \ \ \ ,n \ne 0 \]

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سری دوم : *

\[ \int \sin ax \ dx = -\frac{1}{a} \cos ax \] \[ \int \sin^2 ax\ dx = \frac{x}{2} - \frac{\sin 2ax} {4a} \] \[ \int \sin^3 ax \ dx = -\frac{3 \cos ax}{4a} + \frac{\cos 3ax} {12a} \] \[ \int \sin^n ax \ dx = -\frac{1}{a}{\cos ax} \hspace{2mm}{_2F_1}\left[ \frac{1}{2}, \frac{1-n}{2}, \frac{3}{2}, \cos^2 ax \right] \] \[ \int \cos ax\ dx= \frac{1}{a} \sin ax \] \[ \int \cos^2 ax\ dx = \frac{x}{2}+\frac{ \sin 2ax}{4a} \] \[ \int \cos^3 ax dx = \frac{3 \sin ax}{4a}+\frac{ \sin 3ax}{12a} \] \[ \int \cos^p ax dx = -\frac{1}{a(1+p)}{\cos^{1+p} ax} \times {_2F_1}\left[ \frac{1+p}{2}, \frac{1}{2}, \frac{3+p}{2}, \cos^2 ax \right] \] \[ \int \cos x \sin x\ dx = \frac{1}{2}\sin^2 x + c_1 = -\frac{1}{2} \cos^2x + c_2 = -\frac{1}{4} \cos 2x + c_3 \] \[ \int \cos ax \sin bx\ dx = \frac{\cos[(a-b) x]}{2(a-b)} - \frac{\cos[(a+b)x]}{2(a+b)} , a\ne b \] \[ \int \sin^2 ax \cos bx\ dx = -\frac{\sin[(2a-b)x]}{4(2a-b)} + \frac{\sin bx}{2b} - \frac{\sin[(2a+b)x]}{4(2a+b)} \] \[ \int \sin^2 x \cos x\ dx = \frac{1}{3} \sin^3 x \] \[ \int \cos^2 ax \sin bx\ dx = \frac{\cos[(2a-b)x]}{4(2a-b)} - \frac{\cos bx}{2b} - \frac{\cos[(2a+b)x]}{4(2a+b)} \] \[ \int \cos^2 ax \sin ax\ dx = -\frac{1}{3a}\cos^3{ax} \] \[ \int \sin^2 ax \cos^2 bx dx = \frac{x}{4} -\frac{\sin 2ax}{8a}- \frac{\sin[2(a-b)x]}{16(a-b)} +\frac{\sin 2bx}{8b}- \frac{\sin[2(a+b)x]}{16(a+b)} \] \[ \int \sin^2 ax \cos^2 ax\ dx = \frac{x}{8}-\frac{\sin 4ax}{32a} \] \[ \int \tan ax\ dx = -\frac{1}{a} \ln \cos ax \] \[ \int \tan^2 ax\ dx = -x + \frac{1}{a} \tan ax \] \[ \int \tan^n ax\ dx = \frac{\tan^{n+1} ax }{a(1+n)} \times {_2}F_1\left( \frac{n+1}{2}, 1, \frac{n+3}{2}, -\tan^2 ax \right) \] \[ \int \tan^3 ax dx = \frac{1}{a} \ln \cos ax + \frac{1}{2a}\sec^2 ax \] \[ \int \sec x \ dx = \ln | \sec x + \tan x | = 2 \tanh^{-1} \left (\tan \frac{x}{2} \right) \] \[ \int \sec^2 ax\ dx = \frac{1}{a} \tan ax \] \[ \int \sec^3 x \ {dx} = \frac{1}{2} \sec x \tan x + \frac{1}{2}\ln | \sec x + \tan x | \] \[ \int \sec x \tan x\ dx = \sec x \] \[ \int \sec^2 x \tan x\ dx = \frac{1}{2} \sec^2 x \] \[ \int \sec^n x \tan x \ dx = \frac{1}{n} \sec^n x , n\ne 0 \] \[ \int \csc x\ dx = \ln \left | \tan \frac{x}{2} \right| = \ln | \csc x - \cot x| + C \] \[ \int \csc^2 ax\ dx = -\frac{1}{a} \cot ax \] \[ \int \csc^3 x\ dx = -\frac{1}{2}\cot x \csc x + \frac{1}{2} \ln | \csc x - \cot x | \] \[ \int \csc^nx \cot x\ dx = -\frac{1}{n}\csc^n x, n\ne 0 \] \[ \int \sec x \csc x \ dx = \ln | \tan x | \]

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سایر موارد :

\[ \int {\tan xdx} = \ln \left| {\sec x} \right| + C \] \[ \int {{e^{ax}}} \sin bxdx = {{{e^{ax}}} \over {{a^2} + {b^2}}}(a\;\sin bx - b\cos bx) + C \] \[ \int {{e^{ax}}} \cos bxdx = {{{e^{ax}}} \over {{a^2} + {b^2}}}(a\cos bx + b\sin bx) + C \]