Find ((3θ-2)θ)/(5-²θ-4θ)dθ

MathsCBSE Class 12 · Sample paper, session 2025-26 · 5 marks · Long answer

CBSE Class 12 Mathematics sample paper 2025-26: question 33(B)

From the official CBSE sample question paper for 2025-26, section D, question 33(B), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33(B) · 5 marks · Long answerIntegrals

Find $\int \frac{( 3 s i n θ - 2 ) c o s θ}{5 - c o s ^{2} θ - 4 s i n θ} d θ$

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

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

Hint 1

Change $cos^{2} θ$ into $sin θ$ and notice $cos θ d θ$ in the numerator.

Hint 2

The denominator becomes a perfect square; use partial fractions of the form $\frac{A}{t - 2} + \frac{B}{( t - 2 ) ^{2}}$ .

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.

$I = \int \frac{( 3 sin θ - 2 ) cos θ}{5 - ( 1 - sin ^{2} θ ) - 4 sin θ} d θ$ .
½ mark
Put $sin θ = t$ , $cos θ d θ = d t$ : $I = \int \frac{3 t - 2}{t ^{2} - 4 t + 4} d t = \int \frac{3 t - 2}{( t - 2 ) ^{2}} d t$ .
1 mark
Let $\frac{3 t - 2}{( t - 2 ) ^{2}} = \frac{A}{t - 2} + \frac{B}{( t - 2 ) ^{2}} \Rightarrow 3 t - 2 = A (t - 2) + B$ . Comparing coefficients: $A = 3$ ; $- 2 A + B = - 2 \Rightarrow B = 4$ .
1 mark
$I = \int \frac{3}{t - 2} d t + \int \frac{4}{( t - 2 ) ^{2}} d t = 3 lo g ∣ t - 2∣ - \frac{4}{t - 2} + C$ .
2 marks
$I = 3 lo g ∣ sin θ - 2∣ - \frac{4}{sin θ - 2} + C$ .
½ mark

Answer: $3 lo g ∣ sin θ - 2∣ - \frac{4}{sin θ - 2} + C$

Where the marks are

Writes $cos^{2} θ = 1 - sin^{2} θ$ (½ mark)
Substitutes $t = sin θ$ and gets $\frac{3 t - 2}{( t - 2 ) ^{2}}$ (1 mark)
Partial fractions with $A = 3$ , $B = 4$ (½ + ½ marks)
Integrates both terms correctly (1 + 1 marks)
Final answer in $θ$ (½ mark)

Revise the chapter

Chapter: Integrals (notes, formulas, where marks are lost and every past question)

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 2025-26: every question, by section.

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CBSE Class 12 Mathematics sample paper 2025-26: question 33(B)

Hints

Step-by-step solution

Revise the chapter

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