x-5-e-x-2-dx-Help-please-

Question Number 148408 by cesarL last updated on 27/Jul/21

\int x^{5} e^{x^{2}} d x

H e l p p l e a s e!

Answered by Olaf_Thorendsen last updated on 27/Jul/21

F (x) = \int x^{5} e^{x^{2}} d x

F (x) = \int \frac{1}{2} x^{4} (2 {x e}^{x^{2}}) d x

F (x) = \frac{1}{2} x^{4} e^{x^{2}} - \int 2 x^{3} e^{x^{2}} d x

F (x) = \frac{1}{2} x^{4} e^{x^{2}} - \int x^{2} (2 {x e}^{x^{2}}) d x

F (x) = \frac{1}{2} x^{4} e^{x^{2}} - x^{2} e^{x^{2}} + \int 2 {x e}^{x^{2}} d x

F (x) = \frac{1}{2} x^{4} e^{x^{2}} - x^{2} e^{x^{2}} + e^{x^{2}} + C

F (x) = (\frac{1}{2} x^{4} - x^{2} + 1) e^{x^{2}} + C

Commented by cesarL last updated on 27/Jul/21

t h a n k u s i r

Answered by Math_Freak last updated on 27/Jul/21

\int {(x^{2})}^{2} {x e}^{x^{2}} d x

l e t u = x^{2}

d x = \frac{d u}{2 x}

\int u^{2} {x e}^{u} ∙ \frac{d u}{2 x}

\frac{1}{2} \int u^{2} e^{u} d u

i n t e g r a t i n g b y p a r t s

\frac{1}{2} (u^{2} e^{u} - 2 \int {u e}^{u} d u)

\frac{1}{2} (u^{2} e^{u} - 2 ({u e}^{u} - \int e^{u} d u))

\frac{1}{2} (u^{2} e^{u} - 2 ({u e}^{u} - e^{u})) + C

\frac{u^{2} e^{u}}{2} - {u e}^{u} + e^{u} + C

u = x^{2}

\frac{x^{4} e^{x^{2}}}{2} - x^{2} e^{x^{2}} + e^{x^{2}} + C