Evaluate ( x- x)/(√((2x)))dx

MathsCBSE Class 12 · Sample paper, session 2026-27 · 2 marks · Very short answer

CBSE Class 12 Mathematics sample paper 2026-27: question 21(B)

From the official CBSE sample question paper for 2026-27, section B, question 21(B), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q21(B) · 2 marks · Very short answerIntegrals

Evaluate $\int \frac{c o s x - s i n x}{s i n ( 2 x )} d x$

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2026-27.

Hints

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

Hint 1

The numerator $cos x - sin x$ is the derivative of a simple expression.

Hint 2

Use $(sin x + cos x)^{2} = 1 + sin 2 x$ , then substitute $t = sin x + cos x$ .

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.

Write $sin 2 x = (1 + sin 2 x) - 1 = (sin x + cos x)^{2} - 1$ , so $I = \int \frac{c o s x - s i n x}{( s i n x + c o s x ) ^{2} - 1} d x$ .
½ mark
Put $sin x + cos x = t$ . Then $(cos x - sin x) d x = d t$ .
½ mark
$I = \int \frac{d t}{t ^{2} - 1} = lo g t + t^{2} - 1 + C$
½ mark
$I = lo g sin x + cos x + sin 2 x + C$
½ mark

Answer: $lo g sin x + cos x + sin 2 x + C$

Where the marks are

Expresses $sin 2 x$ as $(sin x + cos x)^{2} - 1$ (½ mark)
Substitutes $t = sin x + cos x$ with $d t = (cos x - sin x) d x$ (½ mark)
Integrates to $lo g ∣ t + t^{2} - 1 ∣$ (½ mark)
Writes the answer back in terms of $x$ (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 2026-27: every question, by section.

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CBSE Class 12 Mathematics sample paper 2026-27: question 21(B)

Hints

Step-by-step solution

Revise the chapter

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