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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.

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

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

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

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

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

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

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

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