# 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.

**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**