Question

Solve: $\int (1+x-x^{-1})e^{x+x^{-1}}dx=$

• A. $(x+1)e^{x+x^{-1}}+C$
• B. $(x-1)e^{x+x^{-1}}+C$
• C. $-x{e}^{x+x^{-1}}+C$
• D. $x{e}^{x+x^{-1}}+C$

Answer 1

The correct option is D $x{e}^{x+x^{-1}}+C$

Let $I=\int (1+x-x^{-1})e^{x+x^{-1}}dx$

$I=\int e^{x+x^{-1}}dx+\int x(1-\frac{1}{x^2})e^{x+x^{-1}}dx$

$I=\int e^{x+x^{-1}}dx+\int {xe}^{x+x^{-1}}d(x+x^{-1})dx$

$I=x{e}^{x+x^{-1}}+C$