Integral Calculus and Differential Equations

10 practice questions on Integral Calculus and Differential Equations for NDA, with answers

Shishya's practice questions, written with AI. Each answer was checked by an automated second pass, not by a person.

1. 1.Evaluate the definite integral: ∫₀¹ (3x² + 2x) dx

   • (A)2
   • (B)3
   • (C)4
   • (D)5
   Show the answer and solution

   Answer: (A) 2

   Solution: Step 1: Find the indefinite integral. ∫(3x² + 2x) dx = 3(x³/3) + 2(x²/2) + C = x³ + x² + C. Step 2: Apply limits from 0 to 1. [x³ + x²]₀¹ = (1³ + 1²) - (0³ + 0²) = (1 + 1) - 0 = 2. Therefore, the answer is 2.

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2. 2.Find ∫ (1/x) dx, where x > 0.

   • (A)log x + C
   • (B)x² + C
   • (C)1/x² + C
   • (D)e^x + C
   Show the answer and solution

   Answer: (A) log x + C

   Solution: Step 1: Recall the standard integral formula. The integral of 1/x with respect to x is log x (natural logarithm) plus a constant of integration. Step 2: Write the answer. ∫(1/x) dx = log x + C, where C is the constant of integration. Therefore, the answer is log x + C.

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3. 3.Evaluate the integral: ∫(3x² + 4x + 5) dx

   • (A)x³ + 2x² + 5x + C
   • (B)x³ + 4x² + 5x + C
   • (C)3x³ + 4x² + 5x + C
   • (D)x³ + 2x² + 5 + C
   Show the answer and solution

   Answer: (A) x³ + 2x² + 5x + C

   Solution: To integrate ∫(3x² + 4x + 5) dx, we apply the power rule to each term separately. The power rule states: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C. For 3x²: ∫3x² dx = 3 × x³/3 = x³. For 4x: ∫4x dx = 4 × x²/2 = 2x². For 5: ∫5 dx = 5x. Combining all terms: ∫(3x² + 4x + 5) dx = x³ + 2x² + 5x + C, where C is the constant of integration.

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4. 4.The value of ∫(0 to π/2) sin x dx is:

   • (A)0
   • (B)1
   • (C)π/2
   • (D)2
   Show the answer and solution

   Answer: (B) 1

   Solution: ∫ sin x dx = -cos x. Evaluating from 0 to π/2: [-cos(π/2)] - [-cos(0)] = 0 - (-1) = 1.

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5. 5.Find the order and degree of the differential equation: d²y/dx² + 3(dy/dx)² = x³

   • (A)Order 2, Degree 1
   • (B)Order 1, Degree 2
   • (C)Order 2, Degree 2
   • (D)Order 3, Degree 1
   Show the answer and solution

   Answer: (A) Order 2, Degree 1

   Solution: The order of a differential equation is the highest derivative present. Here, d²y/dx² is the highest derivative (second derivative), so the order is 2. The degree is the power of the highest order derivative after the equation is made free from radicals and fractions involving derivatives. The highest order derivative d²y/dx² appears to the power 1, so the degree is 1. Note that (dy/dx)² is not the highest order derivative. Therefore, order = 2 and degree = 1.

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6. 6.The value of ∫(3x² + 4x) dx is:

   • (A)x³ + 2x² + C
   • (B)x³ + 4x² + C
   • (C)3x³ + 2x² + C
   • (D)x³/3 + 2x² + C
   Show the answer and solution

   Answer: (A) x³ + 2x² + C

   Solution: ∫(3x² + 4x) dx = 3(x³/3) + 4(x²/2) + C = x³ + 2x² + C.

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7. 7.Solve the differential equation dy/dx = 2x, given that y = 3 when x = 1.

   • (A)y = x² + 2
   • (B)y = x² + 3
   • (C)y = 2x² + 1
   • (D)y = x² + 1
   Show the answer and solution

   Answer: (A) y = x² + 2

   Solution: Step 1: Separate variables and integrate both sides. dy = 2x dx. Integrating: ∫dy = ∫2x dx, which gives y = 2(x²/2) + C = x² + C. Step 2: Use the initial condition y = 3 when x = 1 to find C. Substitute: 3 = 1² + C, so C = 2. Step 3: Write the particular solution. y = x² + 2. Therefore, the answer is y = x² + 2.

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8. 8.The area bounded by the curve y = x², the x-axis, and the lines x = 0 and x = 2 is:

   • (A)4/3 square units
   • (B)8/3 square units
   • (C)2 square units
   • (D)4 square units
   Show the answer and solution

   Answer: (B) 8/3 square units

   Solution: Step 1: The area is given by the definite integral ∫₀² x² dx. Step 2: Find the indefinite integral. ∫x² dx = x³/3 + C. Step 3: Apply limits from 0 to 2. [x³/3]₀² = (2³/3) - (0³/3) = 8/3 - 0 = 8/3. Therefore, the area is 8/3 square units.

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9. 9.Find the area bounded by the curve y = x², the x-axis, and the lines x = 1 and x = 3.

   • (A)26/3 square units
   • (B)8 square units
   • (C)9 square units
   • (D)10 square units
   Show the answer and solution

   Answer: (A) 26/3 square units

   Solution: The area is given by the definite integral: A = ∫₁³ x² dx. Using the power rule: ∫x² dx = x³/3. Evaluating from 1 to 3: A = [x³/3]₁³ = (3³/3) - (1³/3) = 27/3 - 1/3 = 26/3 square units.

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10. 10.If ∫ f(x) dx = x³ - 2x² + 5x + C, then f(x) equals:

    • (A)3x² - 4x + 5
    • (B)x² - 2x + 5
    • (C)3x² - 2x + 5
    • (D)x³ - 2x² + 5
    Show the answer and solution

    Answer: (A) 3x² - 4x + 5

    Solution: Integration and differentiation are inverse operations. If ∫ f(x) dx = x³ - 2x² + 5x + C, then f(x) is the derivative of (x³ - 2x² + 5x + C). Differentiating term by term: d/dx(x³) = 3x², d/dx(-2x²) = -4x, d/dx(5x) = 5, and d/dx(C) = 0. Therefore, f(x) = 3x² - 4x + 5.

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