Q.51 Calculate the following integral (up to two decimal places)
∫01 (x + 3)(x + 1) dx = ________
Step-by-Step Calculation
Expand the integrand: (x + 3)(x + 1) = x2 + x + 3x + 3 = x2 + 4x + 3.
Find the antiderivative: ∫(x2 + 4x + 3) dx = (1/3)x3 + 2x2 + 3x.
Evaluate from 0 to 1: F(1) – F(0) = [(1/3) + 2 + 3] – 0 = 16/3 ≈ 5.333, which rounds to 5.33.
Common Mistakes (Options Explanation)
Users often miscalculate by not expanding properly or evaluation errors.
- Option like 4.00 might come from integrating only x2 + x, ignoring +3: ∫(x2 + x) dx from 0 to 1 = 1/3 + 1/2 = 5/6 ≈ 0.83 (wrong).
- 6.00 could result from upper limit mistake, like F(2) = (8/3) + 8 + 6 = 26/3 ≈ 8.67 (wrong).
- Correct is always 16/3 = 5.33 after rounding.
