We are what we repeatedly do.

Excellence, then, is not an act, but a habit

**Logics for this lecture:**

Question must be in addition/subtraction form.

Try to convert division/multiplication between functions to addition/subtraction by simplification.

If there are functions other than algebraic functions like trigonometry and logarithmic function, then you can use their respective identities to simplify them. Some of the identities are:

\(1+\cos x=2\cos ^{2} \frac{x}{2} \)

\(1-\cos x=2sin ^{2} \frac{x}{2} \)

\(1\pm sin x=\left(\cos \frac{x}{2} \pm sin \frac{x}{2} \right)^{2} \)

\(\frac{1+\tan x}{1-\tan x} =\tan \left(\frac{pi }{4} +x\right)\)

\(\frac{1-\tan x}{1+\tan x} =\tan \left(\frac{pi }{4} -x\right)\)

\(\log \e^{x} =x\)

\(\e^{\log x} =x\)

Questions Discussed in this lecture:

12. \(\int \sin 2x.dx =-\frac{1}{2} \cos 2x+C\)

13. \(\int \frac{1}{\sqrt{x} +\sqrt{x-1} } dx =\frac{2}{3} x^{\frac{3}{2} } -\frac{2}{3} (x-1)^{\frac{3}{2} } +C\)

14. \(\int \left(x-\frac{1}{2} \right) ^{3} dx=\frac{1}{4} \left(x-\frac{1}{2} \right)^{4} +C\)

15. \(\int \frac{e^{x} }{a^{x} } dx=\frac{\left(\frac{e}{a} \right)^{x}}{\log(e/a)} +C\)

16. \(\int \cos ^{2} \alpha dx=x\cos ^{2} \alpha +C\)

17. \(\int \frac{1}{x\cos \alpha +1} dx=\frac{\log |x\cos \alpha +1|}{\cos \alpha } +C\)

18. \(\int \frac{1}{\cos \alpha +x\sin \alpha } dx =\frac{\log |\cos \alpha +x\sin \alpha |}{\sin \alpha } +C\)

19. \(\int \frac{\sin x+\cos x}{\sqrt{1+\sin 2x} } dx =x+C\)

20. \(\int (\sin ^{-1} \sqrt{x} +\cos ^{-1} \sqrt{x} ) dx=\frac{pi x}{2} +C\)

21. \(\int \frac{1}{1-\sin ^{2} x} dx=\tan x+C\)

22. \(\int \frac{2-3\sin x}{\cos ^{2} x} dx=2\tan x-3\sec x+C\)

23. \(\int e^{-\log e^{x} } dx =-e^{-x} +C\)

24. \(\int \sqrt{1+\cos 2x} dx=\sqrt{2} \sin x+C\)

25. \(\int \tan ^{-1} \sqrt{\frac{1-\cos 2x}{1+\cos 2x} } dx =\frac{x^{2} }{2} +C\)

26. \(\int \tan ^{-1} (\sec x+\tan x) dx=\frac{pi }{4} x+\frac{x^{2} }{4} +C\)

27. \(\int \tan ^{-1} \sqrt{\frac{1-\sin x}{1+\sin x} } dx =\frac{pi }{4} x-\frac{x^{2} }{4} +C\)

28. \(\int \left(\sqrt{ax} -\frac{1}{\sqrt{ax} } \right) ^{2} dx=\frac{ax^{2} }{2} +\frac{\log |x|}{a} -2x+C\)

29. \(\int \frac{\sin 4x}{\sin 2x} dx = \sin 2x+C\)

30. \(\int e^{2x+3} dx = \frac{1}{2} e^{2x+3} +C\)

31. \(\int \sec ^{2} (7-4x)dx = -\frac{1}{4} \tan (7-4x)+C \)