The value of d/dx_sin^2x^2sinxe^t^2 dt at x=π is

Question 30 of 31 · Definite Integration

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NIMCET 2025

The value of $\frac{d}{d x} \int_{s in^{2} x}^{2 s in x} e^{t^{2}} d t$ at $x = π$ is

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Answer: Option C — $- 2$

Detailed Solution

Leibniz rule: $\frac{d}{d x} \int_{u}^{v} f (t) d t = f (v) v' - f (u) u'$ .

$v = 2 s in x$ , $v' = 2 cos x$ ; $u = s in^{2} x$ , $u' = 2 s in x cos x$ .

At $x = π$ : $v = 0$ , $v' = - 2$ , $u = 0$ , $u' = 0$ , so the value is $e^{0} (- 2) - e^{0} (0) = - 2$ .

✓ Check

Near

x = π

both limits are close to 0, where

e^{t^{2}} \approx 1

, so the integral is about

2 s in x - s in^{2} x

, whose derivative at

π

2 cos π - s in 2 π = - 2

⚠ Common trap

(A) 2 comes from taking

cos π = 1

instead of

- 1

Quick tip

$f (v) v^{'} - f (u) u^{'} = e^{0} (2 cos π) - e^{0} (sin 2 π) = - 2$

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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.

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