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Provide Solution For  R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise 18.11 Question 4 Maths Textbook Solution.

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Answer:\frac{1}{6} \sec ^{6} x+C

Hint:- Use substitution method to solve this integral.

Given:\int \sec ^{6} x \cdot \tan x d x

Solution:Let,\mathrm{I}=\int \sec ^{6} x \cdot \tan x d x

Re-write,I=\int \sec ^{5} x \sec x \cdot \tan x d x

\mathrm{I}=\int \sec ^{5} x(\sec x \cdot \tan x) d x

Substitute, \sec x=t \Rightarrow \sec x \cdot \tan x d x=d t, then,

I=\int t^{5} \cdot \mathrm{dt} \quad(\text { if }, \sec x=t)

=\frac{t^{5+1}}{5+1}+C        \text { (if, } \left.\int x^{n} d x=\frac{x^{n+1}}{n+1}+c\right)

\left.=\frac{t^{6}}{6}+c=\frac{\sec ^{6} x}{6}+C \quad \text { (if, } \sec x=t\right)

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