Observe the following statements :A: int left | vedclass

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(C) For statement $A$: Let $I = \int \left(\frac{x^2-1}{x^2}\right) e^{\left(\frac{x^2+1}{x}\right)} d x$.We can rewrite the integrand as $I = \int \l

Vedclass · Accepted Answer

$A$ is true,$R$ is false

Vedclass · Answer

(C) For statement $A$: Let $I = \int \left(\frac{x^2-1}{x^2}\right) e^{\left(\frac{x^2+1}{x}\right)} d x$.We can rewrite the integrand as $I = \int \left(1 - \frac{1}{x^2}\right) e^{\left(x + \frac{1}{x}\right)} d x$.Let $t = x + \frac{1}{x}$,then $d t = \left(1 - \frac{1}{x^2}\right) d x$.Substituting these into the integral,we get $I = \int e^t d t = e^t + c = e^{x + \frac{1}{x}} + c = e^{\frac{x^2+1}{x}} + c$.Thus,statement $A$ is true.For statement $R$: The integral $\int f^{\prime}(x) e^{f(x)} d x$ is a standard form. Let $u = f(x)$,then $d u = f^{\prime}(x) d x$.The integral becomes $\int e^u d u = e^u + c = e^{f(x)} + c$.The statement $R$ claims the result is $f(x) + c$,which is incorrect. Therefore,$R$ is false.

Observe the following statements :
$A: \int \left(\frac{x^2-1}{x^2}\right) e^{\frac{x^2+1}{x}} d x = e^{\frac{x^2+1}{x}} + c$
$R: \int f^{\prime}(x) e^{f(x)} d x = f(x) + c$
Then which of the following is true?

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Observe the following statements :$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$$R: \int f^{\prime}(x) e^{f(x)} d x = f(x) + c$Then which of the following is true?