Evaluate (1)/(x√(x^6-1))dx

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

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

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

Q21(A) · 2 marks · Very short answerIntegrals

Evaluate $\int \frac{1}{x x ^{6} - 1} d x$

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

Try to create the derivative of $x^{6}$ in the numerator.

Hint 2

Multiply the numerator and denominator by $x^{5}$ , then substitute $x^{6} - 1 = t^{2}$ .

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.

Multiply the numerator and denominator by $6 x^{5}$ : $I = \int \frac{d x}{x x ^{6} - 1} = \frac{1}{6} \int \frac{6 x ^{5}}{x ^{6} x ^{6} - 1} d x$ . Put $x^{6} - 1 = t^{2}$ , so $6 x^{5} d x = 2 t d t$ and $x^{6} = t^{2} + 1$ .
½ mark
$I = \frac{1}{6} \int \frac{2 t d t}{( t ^{2} + 1 ) t} = \frac{1}{3} \int \frac{d t}{t ^{2} + 1} = \frac{1}{3} tan^{- 1} t + C = \frac{1}{3} tan^{- 1} (x^{6} - 1) + C$
1½ marks

Answer: $\frac{1}{3} tan^{- 1} x^{6} - 1 + C$

Where the marks are

Rewrites the integrand with $6 x^{5}$ and substitutes $x^{6} - 1 = t^{2}$ (½ mark)
Reduces the integral to $\frac{1}{3} \int \frac{d t}{1 + t ^{2}}$ and writes $\frac{1}{3} tan^{- 1} x^{6} - 1 + C$ (1½ marks)

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(A)

Hints

Step-by-step solution

Revise the chapter

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