Observe the following statements : $\mathrm{A}: \int\left(\frac{x^2-1}{x^2}\right) e^{\frac{x^2+1}{x}} d…
Observe the following statements :
$\mathrm{A}: \int\left(\frac{x^2-1}{x^2}\right) e^{\frac{x^2+1}{x}} d x=e^{\frac{x^2+1}{x}}+c$
$\mathrm{R}: \int f^{\prime}(x) e^{f(x)} d x=f(x)+c$
Then which of the following is true ?
Both A and R are true and R is not the correct reason for $\mathrm{A}$
Both A and R are true and R is the correct reason for $A$
$\mathrm{A}$ is true, $\mathrm{R}$ is false
A is false, $R$ is true
Solution
(A) Let $I=\int\left(\frac{x^2-1}{x^2}\right) e^{\left(\frac{x^2+1}{x}\right)} d x$
$=\int\left(1-\frac{1}{x^2}\right) e^{\left(x+\frac{1}{x}\right)} d x$
Let $\quad x+\frac{1}{x}=t$
$\Rightarrow \quad\left(1-\frac{1}{x^2}\right) d x=d t$
$\therefore \quad I=\int e^t d t=e^t+c$
$=e^{x+\frac{1}{x}}+c=e^{\frac{x^2+1}{x}}+c$
(R) Let $I=\int f^{\prime}(x) e^{f(x)} d x$
Let $f(x)=t$ $f^{\prime}(x) d x=d t$
$\therefore \quad I=\int e^t d t=e^t+c$
$=e^{f(x)}+c$
Thus $A$ is true but $R$ is false.