The integral $\int\left(1+x-\frac{1}{x}\right)e^{x+\frac{1}{x}}dx$ is equal to

JEE Main · 2014 · 06 Apr 2014 · Offline · Indefinite Integrals

The integral

\int (1 + x - \frac{1}{x}) e^{x + \frac{1}{x}} d x

is equal to

Answer: (B)

Split the integrand as

\int e^{x + 1/ x} d x + \int x (1 - \frac{1}{x ^{2}}) e^{x + 1/ x} d x

. In the second piece notice that

\frac{d}{d x} (x + \frac{1}{x}) = 1 - \frac{1}{x ^{2}}

, so taking

u = x

and integrating by parts gives

x e^{x + 1/ x} - \int e^{x + 1/ x} d x

. That leftover integral cancels the first piece exactly, leaving

x e^{x + 1/ x} + c

JEE Main · 2014 · 06 Apr 2014 · Offline · Single correct option

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