Помогите решить интеграл
P.S. Желательно запомнить, кроме таблицы интегралов, ещё два часто встречающихся интеграла:
![\int \frac{dx}{sin^2x\cdot cosx}=\int \frac{sin^2x+cos^2x}{sin^2x\cdot cosx}\, dx=\int \frac{sin^2x\, dx}{y\sin^2x\cdot cosx}+\int \frac{cos^2x\, dx}{sin^2x\cdot cosx}=\\\\=\int \frac{dx}{cosx}+\int \frac{cosx\, dx}{sin^2x}=ln|tg(\frac{x}{2}+\frac{\pi}{4})|+\int \frac{d(sinx)}{sin^2x}=ln|tg(\frac{x}{2}+\frac{\pi}{4})|-\frac{1}{sinx}+C\; ;\\\\\\\star \; \; \int \frac{dx}{cosx}=\int \frac{cosx\, dx}{cos^2x}=\int\frac{cosx\, dx}{1-sin^2x}=\int \frac{d(sinx)}{1-sin^2x}=\Big [\; \int \frac{dt}{1-t^2}=\frac{1}{2}\cdot ln|\frac{1+t}{1-t}|+C\; \Big ]=](https://tex.z-dn.net/?f=%5Cint%20%5Cfrac%7Bdx%7D%7Bsin%5E2x%5Ccdot%20cosx%7D%3D%5Cint%20%5Cfrac%7Bsin%5E2x%2Bcos%5E2x%7D%7Bsin%5E2x%5Ccdot%20cosx%7D%5C%2C%20dx%3D%5Cint%20%5Cfrac%7Bsin%5E2x%5C%2C%20dx%7D%7By%5Csin%5E2x%5Ccdot%20cosx%7D%2B%5Cint%20%5Cfrac%7Bcos%5E2x%5C%2C%20dx%7D%7Bsin%5E2x%5Ccdot%20cosx%7D%3D%5C%5C%5C%5C%3D%5Cint%20%5Cfrac%7Bdx%7D%7Bcosx%7D%2B%5Cint%20%5Cfrac%7Bcosx%5C%2C%20dx%7D%7Bsin%5E2x%7D%3Dln%7Ctg%28%5Cfrac%7Bx%7D%7B2%7D%2B%5Cfrac%7B%5Cpi%7D%7B4%7D%29%7C%2B%5Cint%20%5Cfrac%7Bd%28sinx%29%7D%7Bsin%5E2x%7D%3Dln%7Ctg%28%5Cfrac%7Bx%7D%7B2%7D%2B%5Cfrac%7B%5Cpi%7D%7B4%7D%29%7C-%5Cfrac%7B1%7D%7Bsinx%7D%2BC%5C%3B%20%3B%5C%5C%5C%5C%5C%5C%5Cstar%20%5C%3B%20%5C%3B%20%5Cint%20%5Cfrac%7Bdx%7D%7Bcosx%7D%3D%5Cint%20%5Cfrac%7Bcosx%5C%2C%20dx%7D%7Bcos%5E2x%7D%3D%5Cint%5Cfrac%7Bcosx%5C%2C%20dx%7D%7B1-sin%5E2x%7D%3D%5Cint%20%5Cfrac%7Bd%28sinx%29%7D%7B1-sin%5E2x%7D%3D%5CBig%20%5B%5C%3B%20%5Cint%20%5Cfrac%7Bdt%7D%7B1-t%5E2%7D%3D%5Cfrac%7B1%7D%7B2%7D%5Ccdot%20ln%7C%5Cfrac%7B1%2Bt%7D%7B1-t%7D%7C%2BC%5C%3B%20%5CBig%20%5D%3D "\int \frac{dx}{sin^2x\cdot cosx}=\int \frac{sin^2x+cos^2x}{sin^2x\cdot cosx}\, dx=\int \frac{sin^2x\, dx}{y\sin^2x\cdot cosx}+\int \frac{cos^2x\, dx}{sin^2x\cdot cosx}=\\\\=\int \frac{dx}{cosx}+\int \frac{cosx\, dx}{sin^2x}=ln|tg(\frac{x}{2}+\frac{\pi}{4})|+\int \frac{d(sinx)}{sin^2x}=ln|tg(\frac{x}{2}+\frac{\pi}{4})|-\frac{1}{sinx}+C\; ;\\\\\\\star \; \; \int \frac{dx}{cosx}=\int \frac{cosx\, dx}{cos^2x}=\int\frac{cosx\, dx}{1-sin^2x}=\int \frac{d(sinx)}{1-sin^2x}=\Big [\; \int \frac{dt}{1-t^2}=\frac{1}{2}\cdot ln|\frac{1+t}{1-t}|+C\; \Big ]=")
![=\frac{1}{2}\cdot ln\Big |\frac{1+sinx}{1-sinx}\Big |+C=\frac{1}{2}\cdot ln\Big |\frac{1+cos(\frac{\pi}{4}-\frac{x}{2})}{1-cos(\frac{\pi}{4}-\frac{x}{2})}\Big |+C=\\\\=\Big [\; 1+cos\alpha =2cos^2\frac{\alpha }{2}\; ,\; 1-cos\alpha =2sin^2\frac{\alpha }{2}\; \Big ]=\\\\=\frac{1}{2}\cdot ln\Big |\frac{2cos^2(\frac{\pi}{4}-\frac{x}{2})}{2sin^2(\frac{\pi}{4}-\frac{x}{2})}\Big |+C=ln\Big |ctg(\frac{\pi}{4}-\frac{x}{2})\Big |+C=\Big [\; ctg\alpha =tg(\frac{\pi}{2}-\alpha )\; \Big ]=\\\\=ln\Big |tg(\frac{\pi}{2}-(\frac{\pi}{4}-\frac{x}{2}))\Big |+C=ln\Big |tg(\frac{\pi}{4}+\frac{x}{2})\Big |+C\; .](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B2%7D%5Ccdot%20ln%5CBig%20%7C%5Cfrac%7B1%2Bsinx%7D%7B1-sinx%7D%5CBig%20%7C%2BC%3D%5Cfrac%7B1%7D%7B2%7D%5Ccdot%20ln%5CBig%20%7C%5Cfrac%7B1%2Bcos%28%5Cfrac%7B%5Cpi%7D%7B4%7D-%5Cfrac%7Bx%7D%7B2%7D%29%7D%7B1-cos%28%5Cfrac%7B%5Cpi%7D%7B4%7D-%5Cfrac%7Bx%7D%7B2%7D%29%7D%5CBig%20%7C%2BC%3D%5C%5C%5C%5C%3D%5CBig%20%5B%5C%3B%201%2Bcos%5Calpha%20%3D2cos%5E2%5Cfrac%7B%5Calpha%20%7D%7B2%7D%5C%3B%20%2C%5C%3B%201-cos%5Calpha%20%3D2sin%5E2%5Cfrac%7B%5Calpha%20%7D%7B2%7D%5C%3B%20%5CBig%20%5D%3D%5C%5C%5C%5C%3D%5Cfrac%7B1%7D%7B2%7D%5Ccdot%20ln%5CBig%20%7C%5Cfrac%7B2cos%5E2%28%5Cfrac%7B%5Cpi%7D%7B4%7D-%5Cfrac%7Bx%7D%7B2%7D%29%7D%7B2sin%5E2%28%5Cfrac%7B%5Cpi%7D%7B4%7D-%5Cfrac%7Bx%7D%7B2%7D%29%7D%5CBig%20%7C%2BC%3Dln%5CBig%20%7Cctg%28%5Cfrac%7B%5Cpi%7D%7B4%7D-%5Cfrac%7Bx%7D%7B2%7D%29%5CBig%20%7C%2BC%3D%5CBig%20%5B%5C%3B%20ctg%5Calpha%20%3Dtg%28%5Cfrac%7B%5Cpi%7D%7B2%7D-%5Calpha%20%29%5C%3B%20%5CBig%20%5D%3D%5C%5C%5C%5C%3Dln%5CBig%20%7Ctg%28%5Cfrac%7B%5Cpi%7D%7B2%7D-%28%5Cfrac%7B%5Cpi%7D%7B4%7D-%5Cfrac%7Bx%7D%7B2%7D%29%29%5CBig%20%7C%2BC%3Dln%5CBig%20%7Ctg%28%5Cfrac%7B%5Cpi%7D%7B4%7D%2B%5Cfrac%7Bx%7D%7B2%7D%29%5CBig%20%7C%2BC%5C%3B%20. "=\frac{1}{2}\cdot ln\Big |\frac{1+sinx}{1-sinx}\Big |+C=\frac{1}{2}\cdot ln\Big |\frac{1+cos(\frac{\pi}{4}-\frac{x}{2})}{1-cos(\frac{\pi}{4}-\frac{x}{2})}\Big |+C=\\\\=\Big [\; 1+cos\alpha =2cos^2\frac{\alpha }{2}\; ,\; 1-cos\alpha =2sin^2\frac{\alpha }{2}\; \Big ]=\\\\=\frac{1}{2}\cdot ln\Big |\frac{2cos^2(\frac{\pi}{4}-\frac{x}{2})}{2sin^2(\frac{\pi}{4}-\frac{x}{2})}\Big |+C=ln\Big |ctg(\frac{\pi}{4}-\frac{x}{2})\Big |+C=\Big [\; ctg\alpha =tg(\frac{\pi}{2}-\alpha )\; \Big ]=\\\\=ln\Big |tg(\frac{\pi}{2}-(\frac{\pi}{4}-\frac{x}{2}))\Big |+C=ln\Big |tg(\frac{\pi}{4}+\frac{x}{2})\Big |+C\; .")
