CBSE Class 12 Mathematics 2023: question 27

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. Put $\sqrt{x}=t$, so $\frac{1}{2\sqrt{x}}dx=dt$. Then $I=2\int \frac{dt}{(t+1)(t+2)}$.

   ½ mark

2. By partial fractions, $\frac{1}{(t+1)(t+2)}=\frac{1}{t+1}-\frac{1}{t+2}$. So $I=2\int \left(\frac{1}{t+1}-\frac{1}{t+2}\right)dt$.

   1 mark

3. $I=2\left[\log |t+1|-\log |t+2|\right]+C$

   1 mark

4. $I=2\log \left(\frac{\sqrt{x}+1}{\sqrt{x}+2}\right)+C$

   ½ mark

Answer: $2\log \left(\frac{\sqrt{x}+1}{\sqrt{x}+2}\right)+C$

Revise the chapter

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