$\int_{0}^{π} x f (s in x) d x$ is equal to

Answer (Detailed Solution Below)View answer ▾

Answer: Option C — $π \int_{0}^{π /2} f (cos x) d x$

Detailed Solution

Let $I = \int_{0}^{π} x f (s in x) d x$ . Replacing x by $π - x$ : $I = \int_{0}^{π} (π - x) f (s in x) d x$ .

Adding, $2 I = π \int_{0}^{π} f (s in x) d x$ , so $I = \frac{π}{2} \int_{0}^{π} f (s in x) d x = π \int_{0}^{π /2} f (s in x) d x$ .

$\int_{0}^{π /2} f (s in x) d x = \int_{0}^{π /2} f (cos x) d x$ , so $I = π \int_{0}^{π /2} f (cos x) d x$ .

✓ Check

With

f (t) = 1

I = \int_{0}^{π} x d x = \frac{π ^{2}}{2}

, and (C) gives

π \cdot \frac{π}{2} = \frac{π ^{2}}{2}

, while (A) gives

π^{2}

⚠ Common trap

(A) forgets to divide by 2 after adding the two forms of I.

Quick tip

$\int_{0}^{π} x f (sin x) d x = \frac{π}{2} \int_{0}^{π} f (sin x) d x = π \int_{0}^{π /2} f (sin x) d x$

💬 Discussions on this question

Ask a doubt ›

Still stuck on this question? Ask our faculty and fellow students — answers appear here.

Ask a doubt

Definite Integration — Preparation

This question tests your understanding of Definite Integration, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A definite integral gives a number rather than a family of functions. It is evaluated using the fundamental theorem of calculus: the integral of f(x) from a to b equals F(b) - F(a), where F is an antiderivative of f. This topic covers evaluating definite integrals, substitution with a change of limits, integration by parts with limits, and the definite integral as the limit of a sum. The properties of definite integrals are the key to most exam questions. The most useful is the "a + b - x" (or "a - x") property, which turns many difficult integrals into simple ones by adding the integral to itself. Also remember that the integral of an odd function over a symmetric interval from -a to a is zero, while the integral of an even function equals twice the integral from 0 to a. Practise the definite integration questions below, each with a step-by-step solution showing which property is used.

‹ All Definite Integration questions Browse all topics ›

$\int_{0}^{π} x f (s in x) d x$ is equal to

💬 Discussions on this question

More Definite Integration Questions

More Quantitative Aptitude Questions

Definite Integration — Preparation

∫0π​xf(sinx)dx is equal to

💬 Discussions on this question

More Definite Integration Questions

More Quantitative Aptitude Questions

Definite Integration — Preparation

$\int_{0}^{π} x f (s in x) d x$ is equal to