Integrals — Questions & Answers

Sample questions from Integrals

1. Q1. ∫ e^x dx/(1 + e^x)^2 equals:

   • A.ln(1 + e^x) + C
   • B.1/(1 + e^x) + C
   • C.-2/(1 + e^x)^3 + C
   • D.-1/(1 + e^x) + C✓
   Solution

   Put t = 1 + e^x, dt = e^x dx: ∫ dt/t^2 = -1/t = -1/(1 + e^x) + C.

2. Q2. ∫ (2x + 3)/√(x² + 3x + 2) dx equals:

   • A.2√(x²+3x+2) + C✓
   • B.√(x²+3x+2) + C
   • C.ln|x²+3x+2| + C
   • D.(x²+3x+2)^(3/2)/3 + C
   Solution

   A is correct. Let u=x²+3x+2, du=(2x+3)dx. ∫(2x+3)/√(x²+3x+2) dx = ∫du/√u = 2√u + C = 2√(x²+3x+2) + C.

3. Q3. ∫ (ln x)^2 dx equals:

   • A.x(ln x)^2 - 2x ln x - 2x + C
   • B.x(ln x)^2 - 2x ln x + 2x + C✓
   • C.x(ln x)^2 + 2x ln x + 2x + C
   • D.(ln x)^3/3 + C
   Solution

   By parts twice: x(ln x)^2 - 2∫ ln x dx = x(ln x)^2 - 2(x ln x - x) = x(ln x)^2 - 2x ln x + 2x + C.

4. Q4. The property ∫₀^a f(x)dx = ∫₀^a f(a−x)dx is called:

   • A.Symmetry property
   • B.Fundamental Theorem
   • C.Linearity property
   • D.King's property✓
   Solution

   D is correct. This substitution property — replacing x with a−x in a definite integral with limits 0 to a — is commonly called the King's property. It is extremely useful for evaluating definite integrals.

5. Q5. Evaluate ∫ dx/(5 + 4 sin x).

   • A.(2/3) tan^(-1)((4 tan(x/2) + 5)/3) + C
   • B.(2/3) tan^(-1)((5 tan(x/2) + 4)/3) + C✓
   • C.(1/3) tan^(-1)((5 tan(x/2) + 4)/3) + C
   • D.(2/9) tan^(-1)(5 tan(x/2) + 4) + C
   Solution

   With t = tan(x/2) the integral becomes ∫ 2 dt/(5t^2 + 8t + 5) = (2/5)∫ dt/((t + 4/5)^2 + 9/25) = (2/3) tan^(-1)((5t + 4)/3) + C.

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