Find : (x²)/((x²+4)(x²+9)) dx

MathsCBSE Class 12 · Board exam 2024, set 65/1/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2024: question 28(a)

Set in the CBSE Class 12 board exam of 2024, paper 65/1/1, section C, question 28(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28(a) · 3 marks · Short answerIntegrals

Find : $\int \frac{x ^{2}}{( x ^{2} + 4 ) ( x ^{2} + 9 )} d x$

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

Only $x^{2}$ appears, so temporarily treat $x^{2}$ as $t$ for partial fractions.

Hint 2

Find A and B by substituting $t = - 4$ and $t = - 9$ .

Hint 3

Use $\int \frac{d x}{a ^{2} + x ^{2}} = \frac{1}{a} tan^{- 1} \frac{x}{a}$ .

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 $x^{2} = t$ (only for partial fractions): $\frac{t}{( t + 4 ) ( t + 9 )} = \frac{A}{t + 4} + \frac{B}{t + 9}$ , i.e. $t = A (t + 9) + B (t + 4)$ .
½ mark
$t = - 4$ : $- 4 = 5 A \Rightarrow A = - \frac{4}{5}$ ; $t = - 9$ : $- 9 = - 5 B \Rightarrow B = \frac{9}{5}$ . So $I = - \frac{4}{5} \int \frac{d x}{2 ^{2} + x ^{2}} + \frac{9}{5} \int \frac{d x}{3 ^{2} + x ^{2}}$
1½ marks
$I = - \frac{4}{5} \cdot \frac{1}{2} tan^{- 1} \frac{x}{2} + \frac{9}{5} \cdot \frac{1}{3} tan^{- 1} \frac{x}{3} + C = - \frac{2}{5} tan^{- 1} \frac{x}{2} + \frac{3}{5} tan^{- 1} \frac{x}{3} + C$
1 mark

Answer: $- \frac{2}{5} tan^{- 1} \frac{x}{2} + \frac{3}{5} tan^{- 1} \frac{x}{3} + C$

Where the marks are

Puts $x^{2} = t$ and sets up partial fractions (0.5 mark)
Finds A = −4/5, B = 9/5 and splits the integral (1.5 marks)
Integrates to tan⁻¹ terms correctly (1 mark)

Revise the chapter

Chapter: Integrals (notes, formulas, where marks are lost and every past question)
Topic: Integration by Partial Fractions

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 28(a)

Hints

Step-by-step solution

Revise the chapter

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