dxx³(1+x^4)^1/2 equals (A) -(1)/(2x²)√(1+x^4)+c (B) 1/2x√(1+x^4)+c (C) -1/4x√(1+x^4)+c (D) (1)/(4x²)√(1+x^4)+c

-(1)/(2x²)√(1+x^4)+c, option (A)

MathsCBSE Class 12 · Sample paper, session 2024-25 · 1 mark · MCQ

CBSE Class 12 Mathematics sample paper 2024-25: question 12

From the official CBSE sample question paper for 2024-25, section A, question 12, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q12 · 1 mark · MCQIntegrals

$\int \frac{d x}{x ^{3} ( 1 + x ^{4} ) ^{\frac{1}{2}}}$ equals

(A) $- \frac{1}{2 x ^{2}} 1 + x^{4} + c$
(B) $\frac{1}{2 x} 1 + x^{4} + c$
(C) $- \frac{1}{4 x} 1 + x^{4} + c$
(D) $\frac{1}{4 x ^{2}} 1 + x^{4} + c$

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

Take $x^{4}$ out of the square root.

Hint 2

Try $t = 1 + \frac{1}{x ^{4}}$ .

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 integrand as $\frac{1}{x ^{5} ( 1 + x ^{- 4} ) ^{1/2}}$ ; put $t = 1 + x^{- 4}$ , $d t = - 4 x^{- 5} d x$ .
$- \frac{1}{4} \int t^{- 1/2} d t = - \frac{1}{2} t + c = - \frac{1}{2 x ^{2}} 1 + x^{4} + c$ . Option (A).
1 mark

Answer: $- \frac{1}{2 x ^{2}} 1 + x^{4} + c$ , option (A)

Where the marks are

Substitution $t = 1 + x^{- 4}$ and correct integration (1 mark)

Revise the chapter

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

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 2024-25: every question, by section.

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CBSE Class 12 Mathematics sample paper 2024-25: question 12

Hints

Step-by-step solution

Revise the chapter

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