Integration of trigonometric functions $f(\sin{x}, \cos{x})$

$\circ\,\,\text{Universal trigonometric substitution }\,\,t=\tan{\frac{x}{2}}$

$$\sin{x}=\frac{2\tan{\frac{x}{2}}}{1+\tan^2{\frac{x}{2}}}=\frac{2t}{1+t^2}$$ $$\cos{x}=2\cos^2{\frac{x}{2}}-1=\frac{2}{1+\tan^2{\frac{x}{2}}}-1=\frac{1-t^2}{1+t^2}$$ $$dt=\frac{d\frac{x}{2}}{\cos^2{\frac{x}{2}}}\Rightarrow dx=\frac{2dt}{1+t^2}$$

$$\int\frac{dx}{1+5\cos{x}}=2\int\frac{1}{(1+5\frac{1-t^2}{1+t^2})}\frac{dt}{(1+t^2)}=\int\frac{dt}{3-2t^2}=\frac{1}{2\sqrt6}\ln{\Biggl|\frac{2t+\sqrt6}{2t-\sqrt6}\Biggl|}=\frac{1}{2\sqrt6}\ln{\Biggl|\frac{2\tan{\frac{x}{2}}+\sqrt6}{2\tan{\frac{x}{2}}-\sqrt6}\Biggl|}$$

$$\text{I.}\,\,f(-\sin{x}, \cos{x})=-f(\sin{x}, \cos{x})\longrightarrow \cos{x}=t$$ $$\text{II.}\,\,f(\sin{x}, -\cos{x})=-f(\sin{x}, \cos{x})\longrightarrow \sin{x}=t$$ $$\text{III.}\,\,f(-\sin{x}, -\cos{x})=f(\sin{x}, \cos{x})\longrightarrow \tan{x}=t$$

$$\int\frac{\sin{x}dx}{\cos^2{x}-3\cos{x}+2}:\frac{\sin{x}}{\cos^2{x}-3\cos{x}+2}=-\frac{\sin{(-x)}}{\cos^2{x}-3\cos{x}+2}\Rightarrow \cos{x}=t$$ $$I=\int\frac{-dt}{t^2-3t+2}=\int\frac{dt}{\frac{1}{4}-(t-\frac{3}{2})^2}=\ln{\Biggl|\frac{t-1}{t-2}\Biggl|}=\ln{\Biggl|\frac{\cos{x}-1}{\cos{x}-2}\Biggl|}$$

$$\circ\,\,\text{Reducing the degree of}\,\sin{x}\,\,\text{and}\,\,\cos{x}$$

$$\sin^2{x}=\frac{1-\cos{2x}}{2}$$ $$\cos^2{x}=\frac{1+\cos{2x}}{2}$$ $$\sin{\alpha}\sin{\beta}=\frac{1}{2}[\cos{(\alpha-\beta)}-\cos{(\alpha+\beta)}]$$ $$\cos{\alpha}\cos{\beta}=\frac{1}{2}[\cos{(\alpha-\beta)}+\cos{(\alpha+\beta)}]$$ $$\sin{\alpha}\cos{\beta}=\frac{1}{2}[\sin{(\alpha-\beta)}+\sin{(\alpha+\beta)}]$$ $$\circ\,\,\text{Turning the numerator}\,f(x)\,\text{of function}\,\frac{f(\sin{x},\,\cos{x})}{g(\sin{x},\,\cos{x})}\,\text{into a combination like the folowing:}$$ $$\int\frac{a_1\sin{x}+b_1\cos{x}+c_1}{a\sin{x}+b\cos{x}+c}dx=Ax+B\ln{|a\sin{x}+b\cos{x}+c|}+C\int\frac{dx}{a\sin{x}+b\cos{x}+c}$$ $$\int\frac{a_1\sin^2{x}+2b_1\sin{x}\cos{x}+c_1\cos^2{x}}{a\sin{x}+b\cos{x}}dx=A\sin{x}+B\cos{x}+C\int\frac{dx}{a\sin{x}+b\cos{x}}$$ $$\int\frac{a_1\sin{x}+b_1\cos{x}}{a\sin^2{x}+2b\sin{x}\cos{x}+c\cos^2{x}}dx=A\int\frac{dz_1}{\frac{1}{a-\lambda_1}z_1+\lambda_1}+B\int\frac{dz_2}{\frac{1}{a-\lambda_2}z_2+\lambda_2}\quad-\quad\Large{^{z_i=(a-\lambda_i)\sin{x}+b\cos{x}}_{ \lambda_1,\,\lambda_2: (\lambda-a)(\lambda-c)=b^2}}$$ $$\small{*\,A,\,B,\,C\,\text{ are undefined cofficients found by differentiating both parts and equating}}$$

$$\int\cos^5{x}dx$$

$$\int\sin^6{x}dx$$

$$\int\cos^6{x}dx$$

$$\int\sin^2{x}\cos^4{x}dx$$

$$\int\sin^4{x}\cos^5{x}dx$$

$$\int\sin^5{x}\cos^5{x}dx$$

$$\int\sin{5x}\cos{x}dx$$

$$\int\cos{x}\cos{2x}\cos{3x}dx$$

$$\int\sin{x}\sin{\frac{x}{2}}\sin{\frac{x}{3}}dx$$

$$\int\sin{x}\sin{(x+a)}\sin{(x+b)}dx$$

$$\int\cos^2{ax}\cos^2{bx}dx$$

$$\int\sin^3{2x}\cos^2{3x}dx$$

$$\int\frac{1}{1+\cot{x}}dx$$

$$\int\frac{dx}{3+\sin{x}}$$

$$\int\frac{dx}{1+\varepsilon\cos{x}}\,\,\,^{\large{0<\varepsilon<1}}$$

$$\int\frac{dx}{1+\varepsilon \cos{x}}\,\,\,^{\large{\varepsilon>1}}$$

$$\int\frac{dx}{\sin{x}+2\cos{x}+6}$$

$$\int\frac{dx}{2\sin{x}-\cos{x}+5}$$

$$\int\frac{\cos{x}dx}{2-\cos{x}}$$

$$\int\frac{\sin{x}-\cos{x}}{\sin{x}+2\cos{x}}dx$$

$$\int\frac{\sin{x}}{\sin{x}-3\cos{x}}dx$$

$$\int\frac{\sin{x}+2\cos{x}-3}{\sin{x}-2\cos{x}+3}dx$$

$$\int\frac{\sin{x}}{\sqrt2+\sin{x}+\cos{x}}dx$$

$$\int\frac{2\sin{x}+\cos{x}}{3\sin{x}+4\cos{x}-2}dx$$

$$\int\frac{dx}{3+5\tan{x}}$$

$$\int\frac{dx}{(2+\cos{x})\sin{x}}$$

$$\int\frac{dx}{\sin{x}-\sin{a}}$$

$$\int\frac{dx}{\cos{x}+\cos{a}}$$

$$\int\frac{dx}{\sin{(x+a)}\sin{(x+b)}}$$

$$\int\frac{dx}{\sin{(x+a)}\cos{(x+b)}}$$

$$\int\frac{dx}{\cos{(x+a)}\cos{(x+b)}}$$

$$\int\tan{x}\tan{(x+a)}dx$$

$$\int\frac{dx}{\sin^2{x}-\sin{2x}}$$

$$\int\frac{\sin{x}\cos{x}}{\sin{x}+\cos{x}}dx$$

$$\int\frac{dx}{a\sin{x}+b\cos{x}}$$

$$\int\frac{dx}{(a\sin{x}+b\cos{x})^2}$$

$$\int\frac{a_1\sin{x}+b_1\cos{x}}{(a\sin{x}+b\cos{x})^2}dx$$

$$\int\frac{\sin^2{x}}{\sin{x}+2\cos{x}}dx$$

$$\int\frac{dx}{1+\sin^2{x}}$$

$$\int\frac{dx}{1+\cos^2{x}}$$

$$\int\frac{dx}{a^2\sin^2{}+b^2\cos^2{x}}$$

$$\int\frac{\cos^2{x}dx}{(a^2\sin^2{x}+b^2\cos^2{x})^2}$$

$$\int\frac{dx}{(\sin^2{x}+2\cos^2{x})^2}$$

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

$$\int\frac{\sin{x}dx}{\cos^2{x}-3\cos{x}+2}$$

$$\int\frac{\sin{x}dx}{\sin^2{x}-3\sin{x}+2}$$

$$\int\frac{\cos{x}dx}{\cos^2{x}-5\cos{x}+6}$$

$$\int\frac{\sin{x}dx}{\sin^3{x}+\cos^3{x}}$$

$$\int\frac{\sin(2x)dx}{\sin^4{x}+\cos^4{x}}$$

$$\int\frac{\cos(2x)dx}{\sin^4{x}+\cos^4{x}}$$

$$\int\frac{dx}{\sin^4{x}+\cos^4{x}}$$

$$\int\frac{\sin^2{x}\cos{x}}{\sin{x}+\cos{x}}dx$$

$$\int\frac{1}{1+\tan{x}}dx$$

$$\int\frac{\sin{x}}{2\sin{x}+3\cos{x}}dx$$

$$\int\frac{\sin{2x}}{1+\sin^2{x}}dx$$

$$\int\frac{\sin^3{x}}{\cos^4{x}}dx$$

$$\int\frac{\cos^6{x}}{\sin^3{x}}dx$$

$$\int\frac{dx}{\sin^3{x}}$$

$$\int\frac{dx}{\cos^3{x}}$$

$$\int\frac{\sin^4{x}}{\cos^6{x}}dx$$

$$\int\frac{\cos{2x}}{1+\cos^2{x}}dx$$

$$\int\frac{\sin^2{x}\cos^2{x}}{\sin^5{x}+\cos^5{x}}dx$$

$$\int\frac{\cos(2x)\,dx}{\sin^5{x}\cos{x}+\cos^5{x}\sin{x}}$$

$$\int\frac{dx}{\sin^6{x}+\cos^6{x}}$$

$$\int\frac{\cos^2{x}}{\sin{x}+3\cos{x}}dx$$

$$\int\frac{\sin^2{x}}{1+\sin^2{x}}dx$$

$$\int\frac{\cos^2{x}}{1+\cos^2{x}}dx$$

$$\int\frac{\sin{2x}}{1+\cos^4{x}}dx$$

$$\int\frac{\sin{x}\cos{x}}{1+\sin^4{x}}dx$$

$$\int\frac{dx}{\sin^4{x}\cos^4{x}}$$

$$\int\frac{dx}{\sin^3{x}\cos^5{x}}$$

$$\int\frac{\sin^2{x}-\cos^2{x}}{\sin^4{x}+\cos^4{x}}dx$$

$$\int\frac{dx}{\sin{x}\cos^4{x}}$$

$$\int\tan^5{x}dx$$

$$\int\cot^6{x}dx$$

$$\int\frac{2\sin{x}+\cos{x}}{(2\cos{x}-3\sin{x})^2}dx$$

$$\int\frac{1+4\cos^2{x}+2\sin{x}\cos{x}}{3\cos{x}-\sin{x}}dx$$

$$\int\frac{\sin^2{x}-4\sin{x}\cos{x}+3\cos^2{x}}{\sin{x}+\cos{x}}dx$$

$$\int\frac{\sin^2{x}-\sin{x}\cos{x}+2\cos^2{x}}{\sin{x}+2\cos{x}}dx$$

$$\int\frac{2\sin{x}-\cos{x}}{3\sin^2{x}+4\cos^2{x}}dx$$