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JEE Main Mathematics question on Definite Integrals and Areas

The slope of tangent to a curve y=f(x)y = f(x) at [x,f(x)][x, f(x)] is 2x+12x + 1. If the curve passes through the point (1,2)(1, 2), then the area bounded by the curve, the X-axis and the line x=1x = 1 is
  1. A.
    32\dfrac{3}{2}
  2. B.
    43\dfrac{4}{3}
  3. C.
    56\dfrac{5}{6}
  4. D.
    112\dfrac{1}{12}
Show answer and solution

Correct option: C

Solution

dy/dx = 2x + 1, so y = x² + x + C. Using point (1,2): 2 = 1 + 1 + C → C = 0. Thus y = x² + x.

Area = ∫₀¹ (x² + x) dx = [x³/3 + x²/2]₀¹ = 1/3 + 1/2 = 5/6.

Previous-year question, JEE Main.

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