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

CBSE Class 12 Mathematics 2024: question 23(a)

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

Q23(a) · 2 marks · Very short answerIntegrals

Find : $\int x 1 + 2 x d x$

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

Try a substitution that removes the square root.

Hint 2

With $1 + 2 x = t^{2}$ , express both $x$ and $d x$ in terms of $t$ .

Hint 3

Integrate the resulting polynomial and substitute back.

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 $1 + 2 x = t^{2}$ , so $2 d x = 2 t d t \Rightarrow d x = t d t$ and $x = \frac{t ^{2} - 1}{2}$ .
½ mark
$\int x 1 + 2 x d x = \int \frac{t ^{2} - 1}{2} \cdot t \cdot t d t = \frac{1}{2} \int (t^{4} - t^{2}) d t = \frac{1}{2} [\frac{t ^{5}}{5} - \frac{t ^{3}}{3}] + C$
1 mark
$= \frac{( 1 + 2 x ) ^{5/2}}{10} - \frac{( 1 + 2 x ) ^{3/2}}{6} + C$
½ mark

Answer: $\frac{( 1 + 2 x ) ^{5/2}}{10} - \frac{( 1 + 2 x ) ^{3/2}}{6} + C$

Where the marks are

Correct substitution $1 + 2 x = t^{2}$ with $d x = t d t$ (0.5 mark)
Reduces to $\frac{1}{2} \int (t^{4} - t^{2}) d t$ and integrates (1 mark)
Back-substitutes correctly (0.5 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 paper2024 board paper, set 65/1/1: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 12 Mathematics 2024: question 23(a)

Hints

Step-by-step solution

Revise the chapter

More questions from Integrals