First slide

Introduction to integration

Question

$\int (1 + x - x^{- 1}) e^{x + x^{- 1}} dx$ is equal to

Moderate

A

$(x + 1) e^{x + x^{- 1}} + C$
B

$(x - 1) e^{x + x^{- 1}} + C$
C

$- {xe}^{x + x^{- 1}} + C$
D

${xe}^{x + x^{- 1}} + C$

Solution

$\begin{array}{l} \int (1 + x - x^{- 1}) e^{x + x^{- 1}} d x \\ = \int [x e^{x + x^{- 1}} (1 - \frac{1}{x^{2}}) + e^{x + x^{- 1}}] d x \\ = x e^{x + x^{- 1}} - \int e^{x + x^{- 1}} d x + \int e^{x + x^{- 1}} d x \\ = x e^{x + x^{- 1}} + C \end{array}$

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