Evaluate : _0^(π²)/(4) (√(x))/(√(x)) dx

MathsCBSE Class 12 · Board exam 2024, set 65/1/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2024: question 23(b)

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, section B, question 23(b), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q23(b) · 2 marks · Very short answerIntegrals

Evaluate : $\int_{0}^{\frac{π ^{2}}{4}} \frac{s i n x}{x} d x$

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

The expression contains $x$ in two places — substitute for it.

Hint 2

Remember to change the limits: $π^{2} /4 = π /2$ .

Hint 3

After substitution the $t$ cancels, leaving a simple sine integral.

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.

Put $x = t \Rightarrow x = t^{2}, d x = 2 t d t$ . When $x = 0, t = 0$ ; when $x = \frac{π ^{2}}{4}, t = \frac{π}{2}$ .
½ mark
$I = \int_{0}^{π /2} \frac{sin t}{t} \cdot 2 t d t = 2 \int_{0}^{π /2} sin t d t = 2 [- cos t]_{0}^{π /2}$
1 mark
$= 2 (- cos \frac{π}{2} + cos 0) = 2 (0 + 1) = 2$
½ mark

Answer: 2

Where the marks are

Substitution $x = t$ , $d x = 2 t d t$ with changed limits (0.5 mark)
Reduces to $2 \int_{0}^{π /2} sin t d t$ and integrates (1 mark)
Correct value 2 (0.5 mark)

Revise the chapter

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

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 23(b)

Hints

Step-by-step solution

Revise the chapter

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