AP EAMCET · Maths · Indefinite Integration
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 \(A\)
- C \(\mathrm{A}\) is true, \(\mathrm{R}\) is false
- D A is false, \(R\) is true
Answer & Solution
Correct Answer
(C) \(\mathrm{A}\) is true, \(\mathrm{R}\) is false
Step-by-step Solution
Detailed explanation
(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 \(\quad x+\frac{1}{x}=t\) \(\Rightarrow \quad\left(1-\frac{1}{x^2}\right) d x=d t\)…
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