JEE Main Integral Calculus & Differential Equations: Integral Calculus – Chapter Test

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Difficulty: 10 medium · 10 hard

Integral Calculus MCQ questions with answers

The first 3 of 20 questions. Unlock the series for the rest, each with a solution.

1. Q1. Match List I (indefinite integral) with List II (result). List I: A. ∫ dx/(a²-x²); B. ∫ dx/(x²-a²); C. ∫ dx/√(a²-x²); D. ∫ dx/(x²+a²). List II: 1. (1/a) arctan(x/a); 2. arcsin(x/a); 3. (1/2a) ln|(a+x)/(a-x)|; 4. (1/2a) ln|(x-a)/(x+a)|. Choose the correct matching (constants omitted).

   सूची I (अनिश्चित समाकल) का सूची II (परिणाम) से मिलान कीजिए। सूची I: A. ∫ dx/(a²-x²); B. ∫ dx/(x²-a²); C. ∫ dx/√(a²-x²); D. ∫ dx/(x²+a²)। सूची II: 1. (1/a) arctan(x/a); 2. arcsin(x/a); 3. (1/2a) ln|(a+x)/(a-x)|; 4. (1/2a) ln|(x-a)/(x+a)|। सही मिलान चुनिए (अचर छोड़ दिए गए हैं)।

   • A. A-2, B-1, C-3, D-4
   • B. A-3, B-4, C-2, D-1
   • C. A-3, B-4, C-1, D-2
   • D. A-4, B-3, C-2, D-1
   Solution

   These are the standard forms: ∫dx/(a²-x²) = (1/2a)ln|(a+x)/(a-x)|, ∫dx/(x²-a²) = (1/2a)ln|(x-a)/(x+a)|, ∫dx/√(a²-x²) = arcsin(x/a), ∫dx/(x²+a²) = (1/a)arctan(x/a).

   ये मानक रूप हैं: ∫dx/(a²-x²) = (1/2a)ln|(a+x)/(a-x)|, ∫dx/(x²-a²) = (1/2a)ln|(x-a)/(x+a)|, ∫dx/√(a²-x²) = arcsin(x/a), ∫dx/(x²+a²) = (1/a)arctan(x/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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