If x³^4(x^4)(x^4)dx=a^5(x^4)+C, then a is equal to (A) -1/10 (B) 1/20 (C) 1/4 (D) 1/5

MathsCBSE Class 12 · Sample paper, session 2025-26 · 1 mark · MCQ

CBSE Class 12 Mathematics sample paper 2025-26: question 12

From the official CBSE sample question paper for 2025-26, section A, question 12, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q12 · 1 mark · MCQIntegrals

If $\int x^{3} sin^{4} (x^{4}) cos (x^{4}) d x = a sin^{5} (x^{4}) + C$ , then $a$ is equal to

(A) $- \frac{1}{10}$
(B) $\frac{1}{20}$
(C) $\frac{1}{4}$
(D) $\frac{1}{5}$

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

Try the substitution $t = sin (x^{4})$ .

Hint 2

Find $d t$ , then integrate $t^{4}$ with the constant factor.

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.

Put $sin (x^{4}) = t$ . Then $4 x^{3} cos (x^{4}) d x = d t$ , so $x^{3} cos (x^{4}) d x = \frac{1}{4} d t$ .
$\int t^{4} \cdot \frac{1}{4} d t = \frac{t ^{5}}{20} + C = \frac{1}{20} sin^{5} (x^{4}) + C$ , so $a = \frac{1}{20}$ . Option (B).
1 mark

Answer: $a = \frac{1}{20}$ , option (B)

Where the marks are

Substitutes $t = sin (x^{4})$ and integrates to get $a = \frac{1}{20}$ (1 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 12

Hints

Step-by-step solution

Revise the chapter

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