JEE Main 2014 (Offline)Question 1
The integral $\int\left(1+x-\frac1x\right)e^{x+\frac1x}dx$ is equal to
Indefinite Integration
Solution
Concept. The integrand is the product-rule derivative of $x\,e^{x+\frac1x}$. Formulas used. (1) $\frac{d}{dx}e^{x+\frac1x}=\Big(1-\frac1{x^2}\Big)e^{x+\frac1x}$Step 1 - Differentiate $xe^{x+\frac1x}$. By the product rule and (1), $$\frac{d}{dx}\Big(xe^{x+\frac1x}\Big)=e^{x+\frac1x}+x\Big(1-\frac1{x^2}\Big)e^{x+\frac1x}=\Big(1+x-\frac1x\Big)e^{x+\frac1x}.$$ Step 2 - Conclude. This is the integrand, so the integral is $xe^{x+\frac1x}+c$. Answer: (D)