JEE Main Integral Calculus & Differential Equations – Topic Test

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Difficulty: 15 medium · 15 hard

Integral Calculus & Differential Equations MCQ questions with answers

The first 3 of 30 questions. Unlock the series for the rest, each with a solution.

1. Q1. Consider the following statements about continuous functions on [a, b]: 1. If f is continuous on [a, b], then g(x) = ∫_a^x f(t) dt is differentiable with g′(x) = f(x). 2. ∫_a^b f′(x) dx = f(b) − f(a) whenever f′ is continuous on [a, b]. 3. d/dx ∫_a^b f(t) dt = f(b) − f(a). Which of these statements is/are correct?

   [a, b] पर सतत फलनों के बारे में निम्नलिखित कथनों पर विचार कीजिए: 1. यदि f, [a, b] पर सतत है, तो g(x) = ∫_a^x f(t) dt अवकलनीय है जिसमें g′(x) = f(x)। 2. जब भी f′, [a, b] पर सतत हो, ∫_a^b f′(x) dx = f(b) − f(a)। 3. d/dx ∫_a^b f(t) dt = f(b) − f(a)। इनमें से कौन-से कथन सही है/हैं?

   • A. 1 and 2 only / केवल 1 और 2
   • B. 2 and 3 only / केवल 2 और 3
   • C. 1, 2 and 3 / 1, 2 और 3
   • D. 1 and 3 only / केवल 1 और 3
   Solution

   Statements 1 and 2 are the two parts of the Fundamental Theorem of Calculus. Statement 3 is false: ∫_a^b f(t) dt is a constant, so its derivative is 0, not f(b) − f(a).

   कथन 1 और 2 कलन के आधारभूत प्रमेय के दो भाग हैं। कथन 3 असत्य है: ∫_a^b f(t) dt एक अचर है, अतः इसका अवकलज 0 है, न कि f(b) − f(a)।

2. Q2. The value of ∫ dx/(1 + cos x) is

   ∫ dx/(1 + cos x) का मान है

   • A. tan(x/2) + C
   • B. (1/2) tan(x/2) + C
   • C. cot(x/2) + C
   • D. 2 tan(x/2) + C
   Solution

   Using 1 + cos x = 2 cos²(x/2), the integral becomes (1/2)∫ sec²(x/2) dx = tan(x/2) + C.

   1 + cos x = 2 cos²(x/2) का उपयोग करने पर समाकल (1/2)∫ sec²(x/2) dx = tan(x/2) + C बन जाता है।

3. Q3. If F(x) = ∫₀^{x²} √(1 + t³) dt, then F′(x) equals

   यदि F(x) = ∫₀^{x²} √(1 + t³) dt, तो F′(x) बराबर है

   • A. √(1 + x⁶)
   • B. x² √(1 + x⁶)
   • C. 2x √(1 + x⁶)
   • D. 2x √(1 + x³)
   Solution

   By the Leibniz rule, F′(x) = √(1 + (x²)³) · d(x²)/dx = 2x √(1 + x⁶).

   लाइबनिट्स नियम द्वारा, F′(x) = √(1 + (x²)³) · d(x²)/dx = 2x √(1 + x⁶)।

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