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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 18

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Answer: \frac{1}{-3\left ( 5+\tan x \right )^{3}}+c

Hint: You must know about the integral rule of trigonometric functions.

Given: \int \frac{\sec^{2}x}{\left ( 5+\tan x \right )^{4}}dx

Solution:   \frac{\sec^{2}x}{\left ( 5+\tan x \right )^{4}}

t=5+\tan x   differentiate both sides, \left [ \frac{d}{dx}\tan x= \sec^{2} x \right ]


I=\int \frac{\sec^{2}x}{\left ( 5+\tan x \right )^{4}}dx

= \int \frac{dt}{t^{4}}

= \int t^{-4}dt


=\frac{-1}{3t^{3}}+c                                                \left [ \int x^{n} dx= \frac{x^{n+1}}{n+1}\right ]

\therefore I=\frac{1}{-3\left ( 5+\tan x \right )^{3}}+c

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