Q83 / 160 · AP EAMCET 2003 · Mathematics

$\int\left(1+x-x^{-1}\right) e^{x+x^{-1}} d x$ is equal to :

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

Solution

$\int\left(1+x-x^{-1}\right) e^{x+x^{-1}} d x$ $=\int e^{x+x^{-1}} d x+\int\left(x-\frac{1}{x}\right) e^{x+x^{-1}} d x$ $=e^{x+x^{-1}} \int d x-\int\left[\frac{d}{d x}\left(e^{x+x^{-1}}\right)\right] x d x$ $+\int\left(x-\frac{1}{x}\right) e^{x+x^{-1}} d x$ $=x e^{x+x^{-1}}-\int\left(1-\frac{1}{x^2}\right) x e^{x+x^{-1}} d x$ $+\int\left(x-\frac{1}{x}\right) e^{x+x^{-1}} d x$ $=x e^{x+x^{-1}}-\int\left(x-\frac{1}{x}\right) e^{x+x^{-1}} d x$ $+\int\left(x-\frac{1}{x}\right) e^{x+x^{-1}} d x+C$ $=x e^{x+x^{-1}}+C$

Hence, the correct answer is option: 'D'.

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