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Please Solve R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.25 Question 24 Maths Textbook Solution.

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Answer: x \tan \frac{x}{2}+c

Hint: Use trigonometric formulae

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


            \begin{aligned} &\int\left(\frac{x}{1+\cos x}+\frac{\sin x}{1+\cos x}\right) d x \\ &\left.\Rightarrow \int \frac{x}{2 \cos ^{2} \frac{x}{2}}+\frac{2 \sin \frac{x}{2} \cos \frac{x}{2}}{2 \cos ^{2} \frac{x}{2}}\right] d x \\ &\Rightarrow \frac{1}{2} \int x \sec ^{2} \frac{x}{2}+\int \tan \frac{x}{2} d x \\ &\Rightarrow \frac{1}{2}\left[x \frac{\tan \frac{x}{2}}{\frac{1}{2}}-\int 1 \tan \frac{x}{2} \times 2\right]+\int \tan \frac{x}{2} d x \end{aligned}

             \begin{aligned} &\Rightarrow x \tan \frac{x}{2}-\int \tan \frac{x}{2} d x+\int \tan \frac{x}{2} d x \\ &\Rightarrow x \tan \frac{x}{2}+c \end{aligned}


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