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1997

JEE Main 1997 Mathematics question on Definite Integrals and Areas

If g(x)=∫0xcos⁡4t dtg(x) = \int_0^x \cos^4 t \, dt, then g(x+π)g(x + \pi) equals
  1. A.
    g(x)+g(π)g(x) + g(\pi)
  2. B.
    g(x)−g(π)g(x) - g(\pi)
  3. C.
    g(x)g(π)g(x) g(\pi)
  4. D.
    g(x)g(π)\dfrac{g(x)}{g(\pi)}
Show answer and solution

Correct option: A

Solution

Since cos⁴t is periodic with period π, we have g(x+π) = ∫₀^(x+π) cos⁴t dt = ∫₀x cos⁴t dt + ∫_x^(x+π) cos⁴t dt. The second integral equals ∫₀^π cos⁴t dt = g(π) by the periodicity of cos⁴t. Therefore g(x+π) = g(x) + g(π).

[1997]

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