TS 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

Verified Solution

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 \(\quad x+\frac{1}{x}=t\) \(\Rightarrow \quad\left(1-\frac{1}{x^2}\right) d x=d t\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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