If I_1 = -π/4^π/4 (dx)/(1+ 2x) and I_2 = -1/2^1/2 x dx, then show that I_1 - 4I_2 = 0.

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

CBSE Class 12 Mathematics 2026: question 28

Set in the CBSE Class 12 board exam of 2026, paper 65/1/1, section C, question 28, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28 · 3 marks · Short answerIntegrals

If $I_{1} = \int_{- π /4}^{π /4} \frac{d x}{1 + c o s 2 x}$ and $I_{2} = \int_{- 1/2}^{1/2} ∣ x ∣ d x$ , then show that $I_{1} - 4 I_{2} = 0$ .

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

Both integrands are even functions on symmetric limits.

Hint 2

Use $1 + cos 2 x = 2 cos^{2} x$ .

Hint 3

On $[0, \frac{1}{2}]$ , $∣ x ∣ = x$ .

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.

$\frac{1}{1 + cos 2 x}$ is an even function, so
$I_{1} = 2 \int_{0}^{π /4} \frac{d x}{1 + cos 2 x} = 2 \int_{0}^{π /4} \frac{d x}{2 cos ^{2} x} = \int_{0}^{π /4} sec^{2} x d x = [tan x]_{0}^{π /4} = 1$
1½ marks
$∣ x ∣$ is an even function, so $I_{2} = 2 \int_{0}^{1/2} ∣ x ∣ d x = 2 \int_{0}^{1/2} x d x$
½ mark
$= [x^{2}]_{0}^{1/2} = \frac{1}{4}$
½ mark
$I_{1} - 4 I_{2} = 1 - 4 \cdot \frac{1}{4} = 0$ . Hence proved.
½ mark

Answer: $I_{1} = 1$ , $I_{2} = \frac{1}{4}$ , so $I_{1} - 4 I_{2} = 0$

Where the marks are

Uses even-function property and $1 + cos 2 x = 2 cos^{2} x$ to get $I_{1} = 1$ (1½ marks)
Uses even property for $∣ x ∣$ (½ mark)
Evaluates $I_{2} = \frac{1}{4}$ (½ mark)
Shows $I_{1} - 4 I_{2} = 0$ (½ mark)

Revise the chapter

Chapter: Integrals (notes, formulas, where marks are lost and every past question)
Topic: Properties of Definite Integrals

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 28

Hints

Step-by-step solution

Revise the chapter

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