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

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

Answers (1)

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

Hint: Use trigonometric formulae

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

Solution:

            \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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