Evaluate : _0^(π)/(2) 2x ^-1( x) dx

MathsCBSE Class 12 · Board exam 2024, set 65/1/1 · 5 marks · Long answer

CBSE Class 12 Mathematics 2024: question 33(b)

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, section D, question 33(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33(b) · 5 marks · Long answerIntegrals

Evaluate : $\int_{0}^{\frac{π}{2}} sin 2 x tan^{- 1} (sin x) d x$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/1/1.

Hints

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

Hint 1

Write $sin 2 x$ in terms of $sin x$ and $cos x$ , and look for a substitution.

Hint 2

After putting $t = sin x$ , you get a product $t tan^{- 1} t$ , so integrate by parts with $tan^{- 1} t$ as the first function.

Hint 3

Rewrite $\frac{t ^{2}}{1 + t ^{2}}$ as $1 - \frac{1}{1 + t ^{2}}$ before integrating.

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 $I = \int_{0}^{π /2} sin 2 x tan^{- 1} (sin x) d x = \int_{0}^{π /2} 2 sin x cos x tan^{- 1} (sin x) d x$ .
Put $sin x = t \Rightarrow cos x d x = d t$ . When $x = 0$ , $t = 0$ ; when $x = \frac{π}{2}$ , $t = 1$ .
1 mark
$I = 2 \int_{0}^{1} t tan^{- 1} t d t$
1 mark
Integrate by parts, taking $tan^{- 1} t$ as the first function:
$I = 2 [\frac{t ^{2}}{2} tan^{- 1} t - \frac{1}{2} \int \frac{t ^{2}}{1 + t ^{2}} d t]_{0}^{1}$
1½ marks
Since $\frac{t ^{2}}{1 + t ^{2}} = 1 - \frac{1}{1 + t ^{2}}$ :
$I = 2 [\frac{t ^{2}}{2} tan^{- 1} t - \frac{t}{2} + \frac{1}{2} tan^{- 1} t]_{0}^{1}$
1 mark
$I = 2 (\frac{π}{8} - \frac{1}{2} + \frac{π}{8}) = 2 (\frac{π}{4} - \frac{1}{2}) = \frac{π}{2} - 1$
½ mark

Answer: $\frac{π}{2} - 1$

Where the marks are

Writes $sin 2 x = 2 sin x cos x$ , substitutes $sin x = t$ and changes the limits to 0 and 1 (1 mark)
Reduces the integral to $2 \int_{0}^{1} t tan^{- 1} t d t$ (1 mark)
Applies integration by parts correctly (1½ marks)
Integrates $\frac{t ^{2}}{1 + t ^{2}}$ as $t - tan^{- 1} t$ (1 mark)
Substitutes the limits to get $\frac{π}{2} - 1$ (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 paper2024 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 33(b)

Hints

Step-by-step solution

Revise the chapter

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