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Mathematics Exam  >  Mathematics Questions  >  Ifa)3b)√3c)1/3d)none of theseCorrect an... Start Learning for Free
If 
  • a)
    3
  • b)
    √3
  • c)
    1/3
  • d)
    none of these
Correct answer is option 'A'. Can you explain this answer?
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Ifa)3b)√3c)1/3d)none of theseCorrect answer is option 'A'. Can y...
  • Understand the Given Equation:
    We are given the equation:
    Integral from sin(x) to 1 of [t² * f(t)] dt = 1 - sin(x)
    This equation holds true for every x between 0 and π/2.
  • Differentiate Both Sides with Respect to x:
    To find the function f(t), we'll differentiate both sides of the equation with respect to x.
    Let F(x) represent the left side of the equation:
    F(x) = Integral from sin(x) to 1 of [t² * f(t)] dt
    Differentiating F(x) with respect to x using Leibniz's Rule (which allows differentiation under the integral sign), we get:
    F'(x) = -cos(x) * [sin(x)]² * f(sin(x))
    Here, the negative sign comes from differentiating the lower limit sin(x), and cos(x) is the derivative of sin(x).
  • Differentiate the Right Side:
    The right side of the original equation is:
    1 - sin(x)
    Differentiating this with respect to x gives:
    d/dx (1 - sin(x)) = -cos(x)
  • Set the Derivatives Equal:
    Since both sides of the original equation are equal for all x in the given interval, their derivatives must also be equal. Therefore:
    -cos(x) * [sin(x)]² * f(sin(x)) = -cos(x)
    We can simplify this by canceling out the -cos(x) term from both sides (assuming cos(x) is not zero, which it isn't in the interval from 0 to π/2):
    [sin(x)]² * f(sin(x)) = 1
  • Solve for f(sin(x)):
    From the simplified equation:
    [sin(x)]² * f(sin(x)) = 1
    We can solve for f(sin(x)):
    f(sin(x)) = 1 / [sin(x)]²
  • Express f(t) in Terms of t:
    Let t be equal to sin(x). Since x is in the interval from 0 to π/2, t ranges from 0 to 1.
    Therefore, the function f(t) can be expressed as:
    f(t) = 1 / t²
  1. Compute f at 1 divided by the Square Root of 3:
    Finally, we substitute t with 1 divided by the square root of 3:
    f(1/√3) = 1 / (1/√3)²
    Simplifying the denominator:
    (1/√3)² = 1/3
    Therefore:
    f(1/√3) = 1 / (1/3) = 3
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