Lecture 2 Part 5 Integrals Class 12 Maths

Logic for this lecture:

There are a lot of ways to perform substitution in integration.

First way is to substitute the function only if its derivative is also occurring in question. Sometimes, you need to simplify the function to know whether you can apply substitution or not. Play Video

Questions Discussed in this lecture:

41. $\int \frac{\sin x}{\sin 2x} \, \, dx = \frac{1}{2} \log |\sec x+\tan x|+C$

42. $\int _{}^{}\tan ^{4} x\, dx = \frac{1}{3} \tan ^{3} x-\tan x+x+C$

43. $\int \frac{2x+3}{x^{2} +3x} \, dx = \log |x^{2} +3x|+C$

44. $\int \frac{1+\cos x}{x+\sin x} \, dx = \log |x+\sin x|+C$

45. $\int _{}^{}\cos ^{4} 2x\, dx = \frac{3x}{8} +\frac{1}{8} \sin 4x+\frac{1}{64} \sin 8x+C$

46. $\int _{}^{}\frac{\sin ^{2} x}{1+\cos x} \, dx = x-\sin x+C$

47. $\int \frac{\sin ^{6} x}{\cos ^{8} x} \, dx = \frac{\tan ^{7} x}{7} +C$

48. $\int _{}^{}\frac{\cos x-\sin x}{1+\sin 2x} \, dx = -\frac{1}{\cos x+\sin x} \, +C$

49. $\int _{}^{}\tan ^{3} 2x\sec 2x \, dx = \frac{1}{6} \sec ^{3} 2x-\frac{1}{2} \sec 2x+C$

50. $\int \frac{dx}{\cos ^{2} x+\sin 2x} = \frac{1}{2} \log |1+2\tan x|+C$