If ∫ x 5 e - x 2 d x = g x e - x 2 + c , where c is a constant of integration, then g - 1 is equal to

Step-by-step solution

Let I = ∫ x 5 e - x 2 d x Let, - x 2 = t ⇒ x d x = - 1 2 d t ∴   I = - 1 2 ∫ t 2 e t d t By using integrating by parts, i.e. ∫ u · v d x = u ∫ v d x - ∫ d d x u ∫ v d x d x + c , we get I = - 1 2 t 2 ∫ e t d t - ∫ 2 t ∫ e t d t d t + c ⇒ I = - 1 2 t 2 e t - 2 ∫ t e t d t + c Again, applying integrating by parts, we get ⇒ I = - 1 2 t 2 e t - 2 t ∫ e t d t - ∫ 1 · e t d t + c ⇒ I = - 1 2 t 2 e t - 2 t