Evaluate : -1^1 ((2-x)/(2+x) )dx.

MathsCBSE Class 12 · Sample paper, session 2023-24 · 2 marks · Very short answer

CBSE Class 12 Mathematics sample paper 2023-24: question 24

From the official CBSE sample question paper for 2023-24, section B, question 24, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q24 · 2 marks · Very short answerIntegrals

Evaluate : $\int_{- 1}^{1} lo g (\frac{2 - x}{2 + x}) d x$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

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

Hint 1

The limits are symmetric, $- 1$ to $1$ . Check whether the integrand is even or odd.

Hint 2

Replace $x$ by $- x$ and use $lo g \frac{1}{t} = - lo g t$ .

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.

Let $f (x) = lo g (\frac{2 - x}{2 + x})$ . Then $f (- x) = lo g (\frac{2 + x}{2 - x}) = - lo g (\frac{2 - x}{2 + x}) = - f (x)$ , so $f$ is an odd function.
1 mark
For an odd function, $\int_{- a}^{a} f (x) d x = 0$ . Hence $\int_{- 1}^{1} lo g (\frac{2 - x}{2 + x}) d x = 0$ .
1 mark

Answer: $0$

Where the marks are

Shows $f (- x) = - f (x)$ (1 mark)
Uses the odd-function property to get 0 (1 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 paperSample paper 2023-24: every question, by section.

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CBSE Class 12 Mathematics sample paper 2023-24: question 24

Hints

Step-by-step solution

Revise the chapter

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