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NDA 2024 Maths · Mathematics · Definite Integrals · medium

Let F(x)=∫0^xsin (t^2 dt. 1. F'(x)=sin x. 2. F'(0)=0.

Single correct option

Let F(x)=∫0xsin⁡(t2) dtF(x)=\displaystyle\int_0^x\sin(t^2)\,dt. Consider the following statements:
1. F′(x)=sin⁡xF'(x)=\sin x.
2. F′(0)=0F'(0)=0.
Which of the statements given above is/are correct?
  1. A. 1 only
  2. B. 2 only
  3. C. Both 1 and 2
  4. D. Neither 1 nor 2
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Answer: (B) 2 only

By the fundamental theorem of calculus, F′(x)=sin⁡(x2)F'(x)=\sin(x^2), so statement 1 is false because the integrand is evaluated at t=xt=x, giving sin⁡(x2)\sin(x^2) and not sin⁡x\sin x. At x=0x=0, F′(0)=sin⁡0=0F'(0)=\sin0=0, so statement 2 is true. Trap: forgetting to replace tt by xx inside the square.
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