Find : (x²)/((x²+9)(x²+16)) dx

MathsCBSE Class 12 · Board exam 2026, set 65/1/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2026: question 27(b)

Set in the CBSE Class 12 board exam of 2026, paper 65/1/1, section C, question 27(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q27(b) · 3 marks · Short answerIntegrals

Find : $\int \frac{x ^{2}}{( x ^{2} + 9 ) ( x ^{2} + 16 )} d x$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/1/1.

Hints

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

Hint 1

Treat $x^{2}$ as a single variable $t$ to split into partial fractions.

Hint 2

Find A and B by putting $t = - 9$ and $t = - 16$ .

Hint 3

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

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): $\frac{t}{( t + 9 ) ( t + 16 )} = \frac{A}{t + 9} + \frac{B}{t + 16}$ , i.e. $t = A (t + 16) + B (t + 9)$
½ mark
$t = - 9$ : $- 9 = 7 A \Rightarrow A = - \frac{9}{7}$ ; $t = - 16$ : $- 16 = - 7 B \Rightarrow B = \frac{16}{7}$
1 mark
Given integral $= - \frac{9}{7} \int \frac{d x}{x ^{2} + 9} + \frac{16}{7} \int \frac{d x}{x ^{2} + 16}$
½ mark
$= - \frac{9}{7} \cdot \frac{1}{3} tan^{- 1} \frac{x}{3} + \frac{16}{7} \cdot \frac{1}{4} tan^{- 1} \frac{x}{4} + C = - \frac{3}{7} tan^{- 1} \frac{x}{3} + \frac{4}{7} tan^{- 1} \frac{x}{4} + C$
1 mark

Answer: $- \frac{3}{7} tan^{- 1} \frac{x}{3} + \frac{4}{7} tan^{- 1} \frac{x}{4} + C$

Where the marks are

Sets up partial fractions with $x^{2} = t$ (½ mark)
Finds $A = - \frac{9}{7}$ , $B = \frac{16}{7}$ (1 mark)
Rewrites the integral in $x$ (½ mark)
Integrates to the correct $tan^{- 1}$ expression (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 paper2026 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 2026: question 27(b)

Hints

Step-by-step solution

Revise the chapter

More questions from Integrals