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Provide Solution For  R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise 18.22 Question 9 Maths Textbook Solution.

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Answer: \log |2+\tan x|+c

Given:  \int \frac{1}{\cos x(\sin x+2 \cos x)} d x

Hint: Apply substitution method


\begin{aligned} &\int \frac{1}{\cos x(\sin x+2 \cos x)} d x \\ &\int \frac{1}{\cos x \cdot \sin x+2 \cos ^{2} x} d x \end{aligned}

On dividing numerator and denominator by \cos ^{2} x we get

=\int \frac{\sec ^{2} x}{\tan x+2} d x


2+\tan x=t

\sec ^{2} x d x=d t                                                    (Differentiate w.r.t x)

Now,\int \frac{1}{t} d t

\begin{aligned} &=\log |t|+c \\ &=\log |2+\tan x|+c \end{aligned}

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