For any integer n, the value of -π^π e^²x³(2n+1)x dx is (A) -1 (B) 0 (C) 1 (D) 2

MathsCBSE Class 12 · Sample paper, session 2023-24 · 1 mark · MCQ

CBSE Class 12 Mathematics sample paper 2023-24: question 9

From the official CBSE sample question paper for 2023-24, section A, question 9, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q9 · 1 mark · MCQIntegrals

For any integer $n$ , the value of $\int_{- π}^{π} e^{c o s^{2} x} sin^{3} (2 n + 1) x d x$ is

(A)-1
(B)0
(C)1
(D)2

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

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

Hint 1

Check $f (- x)$ .

Hint 2

Use the property $\int_{- a}^{a} f = 0$ for odd $f$ .

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.

Let $f (x) = e^{c o s^{2} x} sin^{3} (2 n + 1) x$ . Then $f (- x) = e^{c o s^{2} x} (- sin (2 n + 1) x)^{3} = - f (x)$ , so $f$ is odd.
The integral of an odd function over $[- π, π]$ is 0: option (B).
1 mark

Answer: 0, option (B)

Where the marks are

Shows the integrand is odd, so the integral is 0 (1 mark)

Revise the chapter

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

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 2023-24: every question, by section.

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CBSE Class 12 Mathematics sample paper 2023-24: question 9

Hints

Step-by-step solution

Revise the chapter

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