Find : (2x²+3)/(x²(x²+9))dx; x≠0.

MathsCBSE Class 12 · Sample paper, session 2023-24 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2023-24: question 26

From the official CBSE sample question paper for 2023-24, section C, question 26, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q26 · 3 marks · Short answerIntegrals

Find : $\int \frac{2 x ^{2} + 3}{x ^{2} ( x ^{2} + 9 )} d x$ ; $x \neq = 0$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

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

Hint 1

The integrand has only even powers of $x$ , so treat $x^{2}$ as a single variable.

Hint 2

After splitting into partial fractions, 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 $x^{2} = t$ (for the partial fractions only): $\frac{2 t + 3}{t ( t + 9 )} = \frac{A}{t} + \frac{B}{t + 9}$ .
½ mark
Then $2 t + 3 = A (t + 9) + B t$ . At $t = 0$ : $A = \frac{1}{3}$ . At $t = - 9$ : $- 15 = - 9 B$ , so $B = \frac{5}{3}$ .
1 mark
So $\int \frac{2 x ^{2} + 3}{x ^{2} ( x ^{2} + 9 )} d x = \frac{1}{3} \int \frac{d x}{x ^{2}} + \frac{5}{3} \int \frac{d x}{x ^{2} + 3 ^{2}}$ .
½ mark
$= - \frac{1}{3 x} + \frac{5}{3} \cdot \frac{1}{3} tan^{- 1} \frac{x}{3} + C = - \frac{1}{3 x} + \frac{5}{9} tan^{- 1} (\frac{x}{3}) + C$ .
1 mark

Answer: $- \frac{1}{3 x} + \frac{5}{9} tan^{- 1} (\frac{x}{3}) + C$

Where the marks are

Substitutes $x^{2} = t$ and sets up partial fractions (½ mark)
Finds $A = \frac{1}{3}$ , $B = \frac{5}{3}$ (1 mark)
Splits the integral correctly (½ mark)
Integrates to $- \frac{1}{3 x} + \frac{5}{9} tan^{- 1} \frac{x}{3} + C$ (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 paperSample paper 2023-24: 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 sample paper 2023-24: question 26

Hints

Step-by-step solution

Revise the chapter

More questions from Integrals