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NDA 2025 Maths · Mathematics · Integrals · medium

Using the substitution u=x^3, the value of ∫0^1x^2e^x^3 dx is

Single correct option

Using the substitution u=x3u=x^3, the value of ∫01x2ex3 dx\displaystyle\int_0^1x^2e^{x^3}\,dx is
  1. A. e−1e-1
  2. B. e−13\dfrac{e-1}3
  3. C. e3\dfrac e3
  4. D. e2−12\dfrac{e^2-1}2
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Answer: (B) e−13\dfrac{e-1}3

Put u=x3u=x^3, du=3x2dxdu=3x^2dx. The integral becomes 13∫01eudu=e−13\frac13\int_0^1e^udu=\frac{e-1}3. The value e−1e-1 forgets the factor 13\frac13, e3\frac e3 forgets to subtract the lower limit contribution e0=1e^0=1, and e2−12\frac{e^2-1}2 belongs to ∫xex2dx\int xe^{x^2}dx type integrals.
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