Calculus, 2ADV C4 EQ-Bank 1

Differentiate \(y=x e^x\). (2 marks)

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Hence, or otherwise, find \(\displaystyle \int_1^e x e^x d x\). (2 marks)
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a. \(\dfrac{d y}{d x}=e^x+x e^x\)

b. \(e^e(e-1)\)

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a. \(y=x e^x\)

\(\dfrac{d y}{d x}=e^x+x e^x\)

b. \(\text{Using part a.}\)

\(\displaystyle\int e^x+x e^x\, d x\)	\(=x e^x+c\)
\(\displaystyle\int_1^e e^x\, d x+\int_1^e x e^x\, d x\)	\(=\left[x e^x\right]_1^e\)

\(\displaystyle\int_1^e x e^x\, d x\)	\(=\left[x e^x\right]_1^e-\left[e^x\right]_1^e\)
	\(=\left[e \cdot e^e-e\right]-\left[e^e-e\right]\)
	\(=e \cdot e^e-e^e\)
	\(=e^e(e-1)\)