Find : (2x+1)/((x+1)²(x-1))dx

MathsCBSE Class 12 · Board exam 2024, set 65/5/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2024: question 28(b)

Set in the CBSE Class 12 board exam of 2024, paper 65/5/1, section C, question 28(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28(b) · 3 marks · Short answerIntegrals

Find : $\int \frac{2 x + 1}{( x + 1 ) ^{2} ( x - 1 )} d x$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/5/1.

Hints

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

Hint 1

A repeated factor $(x + 1)^{2}$ needs terms $\frac{A}{x + 1} + \frac{B}{( x + 1 ) ^{2}}$ .

Hint 2

Put $x = 1$ and $x = - 1$ to get two constants quickly; compare $x^{2}$ coefficients for the third.

Hint 3

$\int \frac{d x}{( x + 1 ) ^{2}} = - \frac{1}{x + 1}$ .

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.

Let $\frac{2 x + 1}{( x + 1 ) ^{2} ( x - 1 )} = \frac{A}{x + 1} + \frac{B}{( x + 1 ) ^{2}} + \frac{C}{x - 1}$ , so $2 x + 1 = A (x + 1) (x - 1) + B (x - 1) + C (x + 1)^{2}$ . Put $x = 1$ : $3 = 4 C \Rightarrow C = \frac{3}{4}$ . Put $x = - 1$ : $- 1 = - 2 B \Rightarrow B = \frac{1}{2}$ . Compare $x^{2}$ : $A + C = 0 \Rightarrow A = - \frac{3}{4}$ . So integrand $= - \frac{3}{4} \cdot \frac{1}{x + 1} + \frac{1}{2} \cdot \frac{1}{( x + 1 ) ^{2}} + \frac{3}{4} \cdot \frac{1}{x - 1}$ .
1½ marks
Integrating: $- \frac{3}{4} lo g ∣ x + 1∣ - \frac{1}{2 ( x + 1 )} + \frac{3}{4} lo g ∣ x - 1∣ + C = \frac{3}{4} lo g \frac{x - 1}{x + 1} - \frac{1}{2 ( x + 1 )} + C$ .
1½ marks

Answer: $\frac{3}{4} lo g \frac{x - 1}{x + 1} - \frac{1}{2 ( x + 1 )} + C$

Where the marks are

Correct partial fractions $A = - \frac{3}{4}$ , $B = \frac{1}{2}$ , $C = \frac{3}{4}$ (1½ marks)
Correct integration of each term (1½ marks)

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 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 28(b)

Hints

Step-by-step solution

Revise the chapter

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