CBSE Class 12 Mathematics 2025: question 28(a)

Hints

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Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

1. Use $\sin x=2\sin \frac{x}{2}\cos \frac{x}{2}$ and $1+\cos x=2{\cos }^2\frac{x}{2}$: $I=\int \frac{x+2\sin \frac{x}{2}\cos \frac{x}{2}}{2{\cos }^2\frac{x}{2}}dx$.

   1 mark

2. Split the integral: $I=\int x\cdot \frac{1}{2}{\sec }^2\frac{x}{2}dx+\int \tan \frac{x}{2}dx$.

   ½ mark

3. Integrate the first part by parts, taking $x$ as the first function. Since $\int \frac{1}{2}{\sec }^2\frac{x}{2}dx=\tan \frac{x}{2}$: $I=x\tan \frac{x}{2}-\int \tan \frac{x}{2}dx+\int \tan \frac{x}{2}dx$.

   1 mark

4. The two remaining integrals cancel, so $I=x\tan \frac{x}{2}+C$.

   ½ mark

Answer: $x\tan \frac{x}{2}+C$

Revise the chapter

• Chapter: Integrals (notes, formulas, where marks are lost and every past question)