# முக்கோணவியல் சார்புகளின் தொகையீடுகளின் பட்டியல்

## சைன் மற்றும் கோசைன் கொண்ட தொகையீடுகள்

: <math>\int\frac{dx}{\cos ax\pm\sin ax} = \frac{1}{a\sqrt{2}}\ln\\tan\(\frac{ax}{2}\pm\frac{\pi}{8}\)\+C</math>  
: <math>\int\frac{dx}{(\cos ax\pm\sin ax)^2} = \frac{1}{2a}\tan\(ax\mp\frac{\pi}{4}\)+C</math>  
: <math>\int\frac{dx}{(\cos x + \sin x)^n} = \frac{1}{n-1}\(\frac{\sin x - \cos x}{(\cos x + \sin x)^{n - 1}} - 2(n - 2)\int\frac{dx}{(\cos x + \sin x)^{n-2}} \)</math>  
: <math>\int\frac{\cos ax\;dx}{\cos ax + \sin ax} = \frac{x}{2} + \frac{1}{2a}\ln\\sin ax + \cos ax\+C</math>  
: <math>\int\frac{\cos ax\;dx}{\cos ax - \sin ax} = \frac{x}{2} - \frac{1}{2a}\ln\\sin ax - \cos ax\+C</math>  
: <math>\int\frac{\sin ax\;dx}{\cos ax + \sin ax} = \frac{x}{2} - \frac{1}{2a}\ln\\sin ax + \cos ax\+C</math>  
: <math>\int\frac{\sin ax\;dx}{\cos ax - \sin ax} = -\frac{x}{2} - \frac{1}{2a}\ln\\sin ax - \cos ax\+C</math>  
: <math>\int\frac{\cos ax\;dx}{\sin ax(1+\cos ax)} = -\frac{1}{4a}\tan^2\frac{ax}{2}+\frac{1}{2a}\ln\\tan\frac{ax}{2}\+C</math>  
: <math>\int\frac{\cos ax\;dx}{\sin ax(1-\cos ax)} = -\frac{1}{4a}\cot^2\frac{ax}{2}-\frac{1}{2a}\ln\\tan\frac{ax}{2}\+C</math>  
: <math>\int\frac{\sin ax\;dx}{\cos ax(1+\sin ax)} = \frac{1}{4a}\cot^2\(\frac{ax}{2}+\frac{\pi}{4}\)+\frac{1}{2a}\ln\\tan\(\frac{ax}{2}+\frac{\pi}{4}\)\+C</math>  
: <math>\int\frac{\sin ax\;dx}{\cos ax(1-\sin ax)} = \frac{1}{4a}\tan^2\(\frac{ax}{2}+\frac{\pi}{4}\)-\frac{1}{2a}\ln\\tan\(\frac{ax}{2}+\frac{\pi}{4}\)\+C</math>  
: <math>\int\sin ax\cos ax\;dx = -\frac{1}{2a}\cos^2 ax +C\,\!</math>  
: <math>\int\sin a_1x\cos a_2x\;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{)}\,\!</math>  
: <math>\int\sin^n ax\cos ax\;dx = \frac{1}{a(n+1)}\sin^{n+1} ax +C\qquad\mbox{(for }n\neq -1\mbox{)}\,\!</math>  
: <math>\int\sin ax\cos^n ax\;dx = -\frac{1}{a(n+1)}\cos^{n+1} ax +C\qquad\mbox{(for }n\neq -1\mbox{)}\,\!</math>  
: <math>\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{)}\,\!</math>  
: also: <math>\int\sin^n ax\cos^m ax\;dx = \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{)}\,\!</math>  
: <math>\int\frac{dx}{\sin ax\cos ax} = \frac{1}{a}\ln\\tan ax\+C</math>  
: <math>\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{)}\,\!</math>  
: <math>\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{)}\,\!</math>  
: <math>\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{)}\,\!</math>  
: <math>\int\frac{\sin^2 ax\;dx}{\cos ax} = -\frac{1}{a}\sin ax+\frac{1}{a}\ln\\tan\(\frac{\pi}{4}+\frac{ax}{2}\)\+C</math>  
: <math>\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{)}\,\!</math>  
: <math>\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{)}\,\!</math>  
: <math>\int\frac{\sin^n ax\;dx}{\cos^m ax} = \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} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!</math>  
: also: <math>\int\frac{\sin^n ax\;dx}{\cos^m ax} = -\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} \qquad\mbox{(for }m\neq n\mbox{)}\,\!</math>  
: also: <math>\int\frac{\sin^n ax\;dx}{\cos^m ax} = \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} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!</math>  
: <math>\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{)}\,\!</math>  
: <math>\int\frac{\cos^2 ax\;dx}{\sin ax} = \frac{1}{a}\(\cos ax+\ln\\tan\frac{ax}{2}\\) +C</math>  
: <math>\int\frac{\cos^2 ax\;dx}{\sin^n ax} = -\frac{1}{n-1}\(\frac{\cos ax}{a\sin^{n-1} ax)}+\int\frac{dx}{\sin^{n-2} ax}\) \qquad\mbox{(for }n\neq 1\mbox{)}</math>  
: <math>\int\frac{\cos^n ax\;dx}{\sin^m ax} = -\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} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!</math>  
: also: <math>\int\frac{\cos^n ax\;dx}{\sin^m ax} = \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} \qquad\mbox{(for }m\neq n\mbox{)}\,\!</math>  
: also: <math>\int\frac{\cos^n ax\;dx}{\sin^m ax} = -\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} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!</math>