NIMCET 2025 — Mathematics PYQ

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Question

NIMCET 2025 — Mathematics PYQ

NIMCET | Mathematics | 2025

$Find the value of \frac{d}{d x} (\int_{s i n x^{2}}^{2 s i n x} e^{t^{2}} d t) at x = π$

Choose the correct answer:

Explanation

Formula (Leibniz Rule):

$\frac{d}{d x} (\int_{Lower}^{Upper} f (t) d t) = f (Upper) \cdot \frac{d}{d x} (Upper) - f (Lower) \cdot \frac{d}{d x} (Lower)$

Step-by-Step Calculation:

Step 1: Upper limit ka derivative nikalo

$Upper = 2 sin x$
$\frac{d}{d x} (2 sin x) = 2 cos x$

Step 2: Lower limit ka derivative nikalo

$Lower = sin x^{2}$
$\frac{d}{d x} (sin x^{2}) = cos (x^{2}) \cdot 2 x = 2 x cos x^{2}$

Step 3: Formula me values put karo

$\frac{d}{d x} I (x) = [e^{(2 s i n x)^{2}} \cdot 2 cos x] - [e^{(s i n x^{2})^{2}} \cdot 2 x cos x^{2}]$

Step 4: $x = π$ put karo

Hum jaante hain ki:

$sin π = 0$
$cos π = - 1$

Ab bas value rakh do:

$Value = [e^{(2 \cdot 0)^{2}} \cdot 2 (- 1)] - [e^{(s i n π^{2})^{2}} \cdot 2 π cos (π^{2})]$

$Value = [e^{0} \cdot (- 2)] - [Second Term]$

$Value = 1 \cdot (- 2) - [Second Term]$

$Value = - 2 - 2 π cos (π^{2}) e^{s i n^{2} (π^{2})}$

Choose the correct answer:

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