Find √((x+2)/(x-2)) dx

MathsCBSE Class 12 · Board exam 2026, set 65/1/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2026: question 27(a)

Set in the CBSE Class 12 board exam of 2026, paper 65/1/1, section C, question 27(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q27(a) · 3 marks · Short answerIntegrals

Find $\int \frac{x + 2}{x - 2} d x$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Multiply numerator and denominator inside the root by $x + 2$ .

Hint 2

Split $\frac{x + 2}{x ^{2} - 4}$ into $\frac{x}{x ^{2} - 4} + \frac{2}{x ^{2} - 4}$ .

Hint 3

Use substitution for the first and the standard formula $\int \frac{d x}{x ^{2} - a ^{2}}$ for the second.

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.

Rationalise: $I = \int \frac{x + 2}{x - 2} \cdot \frac{x + 2}{x + 2} d x = \int \frac{x + 2}{x ^{2} - 4} d x$
1 mark
$I = \frac{1}{2} \int \frac{2 x}{x ^{2} - 4} d x + 2 \int \frac{1}{x ^{2} - 4} d x = I_{1} + I_{2}$
1 mark
For $I_{1}$ put $x^{2} - 4 = t \Rightarrow 2 x d x = d t$ : $I_{1} = \frac{1}{2} \int \frac{d t}{t} = t = x^{2} - 4$
½ mark
$I_{2} = 2 lo g x + x^{2} - 4$
$∴ I = x^{2} - 4 + 2 lo g x + x^{2} - 4 + C$
½ mark

Answer: $x^{2} - 4 + 2 lo g x + x^{2} - 4 + C$

Where the marks are

Rationalises to $\int \frac{x + 2}{x ^{2} - 4} d x$ (1 mark)
Splits into two standard integrals (1 mark)
Evaluates $I_{1} = x^{2} - 4$ (½ mark)
Evaluates $I_{2} = 2 lo g ∣ x + x^{2} - 4 ∣$ and writes final answer with $C$ (½ mark)

Revise the chapter

Chapter: Integrals (notes, formulas, where marks are lost and every past question)
Topic: Integration by Substitution

Practise this chapterBoard-style questions from Integrals, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2026: question 27(a)

Hints

Step-by-step solution

Revise the chapter

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