If f(a+b-x)=f(x), then _a^b x f(x)dx is equal to (A) (a+b)/(2)_a^b f(b-x)dx (B) (a+b)/(2)_a^b f(a-x)dx (C) (b-a)/(2)_a^b f(x)dx (D) (a+b)/(2)_a^b f(x)dx

(a+b)/(2)_a^b f(x) dx, option (D)

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

CBSE Class 12 Mathematics sample paper 2025-26: question 11

From the official CBSE sample question paper for 2025-26, section A, question 11, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q11 · 1 mark · MCQIntegrals

If $f (a + b - x) = f (x)$ , then $\int_{a}^{b} x f (x) d x$ is equal to

(A) $\frac{a + b}{2} \int_{a}^{b} f (b - x) d x$
(B) $\frac{a + b}{2} \int_{a}^{b} f (a - x) d x$
(C) $\frac{b - a}{2} \int_{a}^{b} f (x) d x$
(D) $\frac{a + b}{2} \int_{a}^{b} f (x) d x$

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

Replace $x$ by $a + b - x$ inside the integral.

Hint 2

Add the new form of $I$ to the original $I$ .

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 $I = \int_{a}^{b} x f (x) d x$ . By the property $\int_{a}^{b} g (x) d x = \int_{a}^{b} g (a + b - x) d x$ , $I = \int_{a}^{b} (a + b - x) f (a + b - x) d x = \int_{a}^{b} (a + b - x) f (x) d x$ [given $f (a + b - x) = f (x)$ ].
So $I = (a + b) \int_{a}^{b} f (x) d x - I$ , giving $2 I = (a + b) \int_{a}^{b} f (x) d x$ and $I = \frac{a + b}{2} \int_{a}^{b} f (x) d x$ . Option (D).
1 mark

Answer: $\frac{a + b}{2} \int_{a}^{b} f (x) d x$ , option (D)

Where the marks are

Applies the $a + b - x$ property and solves for $I$ (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 2025-26: every question, by section.

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CBSE Class 12 Mathematics sample paper 2025-26: question 11

Hints

Step-by-step solution

Revise the chapter

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