Integral Calculus
Definite Integrals, Power Rule
jee_main_2026_april_4_shift_2
Grade 12
Question:
If ∫_0^1 (x^2 + 2x) dx = a, and ∫_0^1 (x^3 + 2x^2) dx = b, find the value of 3a - 2b.
A. 1
B. 3/2
C. 13/6
D. 5/2
Step-by-Step Solution
Key Concept: Evaluate both definite integrals using power rule, then compute the expression.
Step 1: a = ∫_0^1 (x^2 + 2x) dx = [x^3/3 + x^2]_0^1 = 1/3 + 1 = 4/3. Step 2: b = ∫_0^1 (x^3 + 2x^2) dx = [x^4/4 + 2x^3/3]_0^1 = 1/4 + 2/3 = (3+8)/12 = 11/12. Step 3: 3a - 2b = 3(4/3) - 2(11/12) = 4 - 22/12 = 4 - 11/6 = (24-11)/6 = 13/6.
Correct Answer: C
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