∫_2 ^4 x^3 e^x dx


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Question Number 118753 by carlosmald last updated on 19/Oct/20

$\int_{2}^{4} x^{3} e^{x} d x$
Commented by mohammad17 last updated on 19/Oct/20

$\int_{2}^{4} x^{3} e^{x} d x = \int_{2}^{4} (x^{3} e^{x} + 3 x^{2} e^{x} - 3 x^{2} e^{x}) d x$ $= \int_{2}^{4} d (x^{3} e^{x}) - 3 \int_{2}^{4} (x^{2} e^{x} + 2 {x e}^{x} - 2 {x e}^{x}) d x$ $= \int_{2}^{4} d (x^{3} e^{x}) - 3 \int_{2}^{4} d (x^{2} e^{x}) + 6 \int_{2}^{4} ({x e}^{x} + e^{x} - e^{x}) d x$ $= \int_{2}^{4} d (x^{3} e^{x}) - 3 \int_{2}^{4} d (x^{2} e^{x}) + 6 \int_{2}^{4} d ({x e}^{x}) - 6 \int_{2}^{4} e^{x} d x$ $= {(x^{3} e^{x} - 3 x^{2} e^{x} + 6 {x e}^{x} - 6 e^{x})}_{2}^{4}$ $= (64 e^{4} - 48 e^{4} + 24 e^{4} - 6 e^{4}) - (8 e^{2} - 12 e^{2} + 12 e^{2} - 6 e^{2})$ $= 34 e^{4} - 2 e^{2}$ $b y : M o h a m m a d A l - d o l a i m y$
	Answered by bemath last updated on 19/Oct/20

	$b y T a n z a l i n f o r m u l a$ $\| \begin{matrix} x^{3} e^{x} \\ 3 x^{2} e^{x} (+) \\ 6 x e^{x} (-) \\ 6 e^{x} (+) \\ 0 e^{x} (-) \end{matrix} \|$ $\int_{2}^{3} x^{3} e^{x} d x = {[x^{3} e^{x} - 3 x^{2} e^{x} + 6 {x e}^{x} - 6 e^{x}]}_{2}^{3}$ $= {(e^{x} [x^{3} - 3 x^{2} + 6 x - 6])}_{2}^{3}$ $= 12 e^{3} - 2 e^{2}$
	Answered by Dwaipayan Shikari last updated on 19/Oct/20

	$\int x^{3} e^{x} d x = x^{3} e^{x} - 3 \int x^{2} e^{x}$ $= x^{3} e^{x} - 3 x^{2} e^{x} + 6 \int {x e}^{x}$ $= x^{3} e^{x} - 3 x^{2} e^{x} + 6 {x e}^{x} - 6 e^{x}$ $= e^{x} (x^{3} - 3 x^{2} + 6 x - 6) + C$ $\int_{2}^{4} x^{3} e^{x} = 34 e^{4} - 2 e^{2}$
	Answered by peter frank last updated on 19/Oct/20

	$by part$
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