Verified Solution • 2025–26 Session moderate

QJEEM19Ma92a4: JEE Main 2019 Mathematics - Class 12 JEEM19Ma92a4 Solution: If , where c is a constant of integration, then (–1) is equal…

If ∫x5e−x2dx=g(x)e−x2+c\int {{x^5}} {e^{ - {x^2}}}dx = g\left( x \right){e^{ - {x^2}}} + c , where c is a constant of integration, then gg (–1) is equal to :

Multiple Choice Options

  • A 1
  • B -1
  • C −52- {5 \over 2}
  • D −12- {1 \over 2}

Step-by-Step Verified Solution

Let x2 = t

⇒12∫t2e−tdt\Rightarrow {1 \over 2}\int {{t^2}{e^{ - t}}dt}

⇒12[−t2e−t+∫2te−tdt]\Rightarrow {1 \over 2}\left[ { - {t^2}{e^{ - t}} + \int {2t{e^{ - t}}dt} } \right]

⇒−t2e−t2−te−t−e−t\Rightarrow {{ - {t^2}{e^{ - t}}} \over 2} - t{e^{ - t}} - {e^{ - t}}

⇒(−x42−x2−1)e−x2+c\Rightarrow \left( { - {{{x^4}} \over 2} - {x^2} - 1} \right){e^{ - {x^2}}} + c

Then g(x)=−x42−x2−1g(x) = - {{{x^4}} \over 2} - {x^2} - 1

⇒g(−1)=−12−1−1\Rightarrow g( - 1) = - {1 \over 2} - 1 - 1

⇒−52\Rightarrow - {5 \over 2}

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