# Lecture 2 Part 8 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:**

**59. \int{\frac{{cos}^5{x}}{sin{x}}dx} = log{|}sin{x}|+\frac{1}{4}{sin}^4{x}-{sin}^2{x}+C **

*60. \int\frac{4x+1}{\sqrt{2x^2+x-3}}dx=2\sqrt{2x^2+x-3}+C *

*61. \int{\frac{dx}{e^x+e^{-x}}={tan}^{-1}{(}e^x)+C} *

*62. \int{cos{6}x\sqrt{1+sin{6}x}dx=\frac{1}{9}(1+sin{6}x)^\frac{3}{2}+C} *

*63. \int{{cos}^2{x}.e^{logsin{x}}dx=-\frac{1}{4}{cos}^4{x}}+C *

*64. \int{\frac{\sqrt{tan{x}}}{sin{2}x}dx}=2\sqrt{tan{x}}+C *

*65. \int\frac{{cos}^9{x}}{sin{x}}dx=log{|}sin{x}|-2{sin}^2{x}+\frac{3}{2}{sin}^4{x}-\frac{2}{3}{sin}^6{x}+\frac{1}{8}{sin}^8{x}+C *

*66. \int{e^{3log{x}}(x^4+1)^{-1}dx=\frac{1}{4}log{(}x^4+1)+C} *

*67. \int_{0}^{\frac{\pi}{4}}{{sin}^3{2}tcos{2}tdt=\frac{1}{8}} *

*69. \int_{4}^{9}{\frac{\sqrt x}{(30-x^\frac{3}{2})^2}dx}=\frac{19}{99} *