# Lecture 2 Part 7 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:**

**51. \int\frac{cos{2}x}{(cos{x}+sin{x})^2}dx = \cos{x}+sin{x}|+C **

**52. \int\frac{dx}{sin{x}{cos}^3{x}} = {|}tan{x}|+\frac{1}{2}{tan}^2{x}+C **

**53. \int\frac{x^2+4x}{x^3+6x^2+5}dx = \frac{1}{3}log{|}x^3+6x^2+5|+C **

**54. \int{sec{x}.log{(}sec{x}+tan{x})dx}\ =\ \frac{(log{|}sec{x}+tan{x}|)^2}{2}+C **

**55. \int{cot{x}.logsin{x}dx} =\frac{(logsin{x})^2}{2}+C **

**56. \int{{cot}^3{x}{cosec}^4{x}dx} = -\left(\frac{{cot}^6{x}}{6}+\frac{{cot}^4{x}}{4}\right)+C **

**57. \int\frac{x^{e-1}+e^{x-1}}{x^e+e^x}dx = \frac{1}{e}log{|}x^e+e^x|+C **

**58. \int{\frac{e^x(1+x)}{{cos}^2{(}e^xx)}dx = {\tan (}xe^x)+C} **