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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefiite Integrals Excercise Fill in the Blanks Question 3

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\Rightarrow k=-\frac{1}{4}


Solve this integration and then compare with R.H.S.

\begin{aligned} &I=\int \frac{1}{\tan x+\cot x} d x \\ &I=\int \frac{1}{\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}} d x \\ &I=\int \frac{1}{\frac{\sin ^{2} x+\cos ^{2} x}{\cos x \sin x}} d x \end{aligned}
\begin{aligned} &I=\int \frac{\cos x \sin x}{\sin ^{2} x+\cos ^{2} x} d x \quad\quad\quad\quad\quad\quad\quad\quad\left[\cos ^{2} x+\sin ^{2} x=1\right] \\ &I=\int \sin x \cos x d x \end{aligned}

Multiply and divide by 2
\begin{aligned} &I=\int \frac{2 \sin x \cos x}{2} d x \\ &I=\int \frac{\sin 2 x}{2} d x \quad \quad[\sin 2 x=2 \sin x \cos x] \end{aligned}
\begin{aligned} &I=-\frac{\cos 2 x}{4}+c \\ &=k \cos 2 x+c \\ &\Rightarrow k=-\frac{1}{4} \end{aligned}


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