Q114 / 160 · AP EAMCET 2006 · Mathematics

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 ?

  • A
    Both A and R are true and R is not the correct reason for $\mathrm{A}$
  • B
    Both A and R are true and R is the correct reason for $A$
  • Correct answer
    C
    $\mathrm{A}$ is true, $\mathrm{R}$ is false
  • D
    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.

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

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